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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I ■«WK^Pi^==- TTvTT!? n^/^'^T'^N TEXT-BOOKS OF SCIENCE ADATTSO FOB THB USB OP TISANS AND STUDENTS IN PUBLIC AND SCIENCE SCHOOLS PRACTICAL PHYSICS Q e 1 Si : = ®1 o ®l VaUllBRS FOR KUDING LBNGTKS AND ANQLES. -PRACTICAL PHYSICS ' ~1^^ BY R. T. GLAZEBROOK, M.A., F.R.S. DIWKTOK OF THB NATIONAL PHYSICAL LABORATORY AKD W. N. SHAW, M.A., F.R.S. PBLLOW OP BMMANUKL COLLBGB NEW EDITION Of r;-.'-. ^ \ UNIVE.vi FY OF LONGMANS, GREEN, AND CO. 39 PATERNOSTER ROW, LONDON NEW YORK AND BOMBAY 1905 All rights reservtd Cdh 1 Cj (o rt^-^^' BIBLIOGRAPHICAL NOTE, First printed January 1885 ; Reprinted May 1886, December 1888. Rei'ised Edition February 1893 ; Reprinted April iS^, January 1 899, Ncvember i^oo, January 1 902, and January 1904. New Edition September 1905. >< PREFACE TO THE FOURTH EDITION. Ths issae of a new edition affords us ttie opportunity of making some alterations and additions which the experience of ourselves or our successors at the Cavendish Laboratory has shewn to be desirable. The development of phjrsical science on the lines indicated by the principle of the conservation of energy has made more conspicuous the importance of experimental Dynamics as the basis of experimental physics, so that some considerable space has been given to that branch of the subject, and a good deal of attention has been devoted to the geometrical representation of rates of variation, espe- cially as illustrating the determination of the velocity and acceleration of a body the position of which is known foi successive instants of time. Geometrical representation has, indeed, been kept in view throughout. The advances that have been made in the sciences of magnetism and electro-magnetism have also necessitated lome considerable additions. The chapter on magnetism i: 7' .1 viH Preface. has been enlarged, and a chapter on electro-magnetic induction has been added. It has been thought better not to disturb the numbering of the sections, and the new sections have therefore been separately numbered A-Z and r to 9. In the preparation of this edition we are greatly mdebted both to Mr. H. F. Newall, who was demonstrator when the apparatus for many of the new sections was first set up, and also especially to Mr. G. F. C. Searle, of Peterhouse, upon whose version of the Laboratory MSS. the text of many of the new sections depends. Mr. Searl^ besides contributing the section on the djmamical equiva- lent of heat, has abo been good enough to revise the whole of the proof sheets and to give us the advantage of his experience in the Laboratory by making numerous valuable suggestions. Many of the original drawings for the figures were made for us by Mr. Hayles, the Lecture Assistant at the Laboratory. R. T. GLAZEBROOK. W. N. SHAW. J^anuary 6, i9q% PREFACE. This book is intended for the assistance of Students and Teachers in Physical Laboratories. The absence of any lx>ok covering the same ground made it necessary for us, in conducting the large elementary classes in Practical Physics at the Cavendish Laboratory, to write out in MS. books the practical details of the different experiments. The increase in the number of well-equipped Physical Laboratories has doubtless placed many teachers in the same position as we ourselves were in before these books were compiled ; we have therefore collected together the manuscript notes in the present volume, and have added such general explana- tions as seemed necessary. In offering these descriptions of experiments for publica- tion we are met at the outset by a difficulty which may prove serious. The descriptions, in order to be precise, must refer to particular forms of instruments, and may there- fore be to a certain extent inapplicable to other instruments of the same kind but with some difference, perhaps in the arrangement for adjustment, perhaps in the method of graduation. Spherometers, spectrometers, and katheto- meters are instruments with which this difficulty is particu- larly likely to occur. With considerable diffidence we have thought it best to adhere to the precise descriptions referrir X Preface. to instruments in use in our own Laboratory, trusting that the necessity for adaptation to corresponding instruments used elsewhere will not seriously impair the usefulness of the book. Many of the experiments, however, which we have selected for description require only very simple apparatus, a good deal of which has in our case been constructed in the Laboratory itselC We owe much to Mr. G. Gordon, the Mechanical Assistant at the Cavendish Laboratory, for his ingenuity and skill in this respect Our general aim in the book has been to place before the reader a description of a course of experiments which shall not only enable him to obtain a practical acquaintance with methods of measurement, but also as far as possible illustrate the more important principles of the various sub- jects. We have not as a rule attempted verbal explanations of the principles, but have trusted to the ordinary physical text-books to supply the theoretical parts necessary for understanding the subject ; but whenever we have not been able to call to mind passages in the text-books sufficiently explicit to serve as introductions to the actual measurements, we have either given references to standard works or have endeavoured to supply the necessary information, so that a student might not be asked to attempt an experiment without at least being in a position to find a satisfoctory explanation of its method and principles. In following out this plan we have found it necessary to interpolate a considerable amount of more theoretical information. The theory of the balance has been given in a more complete form 'than is usual in mechanical text-books ; the introductions to the measure- ment of fluid pressure, thermometry, and calorimetry have been inserted in order to accentuate certain important prac- tical points which, as a rule, are only briefly touched upon ; Preface, xi while the chapter on h3rgrometr7 is intended as a complete elementary account of the subject We have, moreover, found it necessary to adopt an entirely different style in those chapters which treat of magnetism and electricity. These subjects» r^^arded from the point of view of the practical measurement of magnetic and electric quantities, present a somewhat different aspect from that generally takoL We have accordingly given an outline of the general theory of these subjects as developed on the lines indicated by the electro-magnetic system of measurement, and the arrangement of the experiments is intended, as far as possi- ble, to illustrate the successive steps in the development The. limits of the space at our disposal have compelled OS to be as concise as possible ; we have, therefore, been unable to illustrate the theory as amply as we could have wished. We hope, however, that we have been suc- cessful in the endeavour to avoid sacrificing clearness to brevity. We have made no attempt to give anything like a com- plete list of the experiments that may be performed with the apparatus that is at the present day regarded as the ordinary equipment of a Physical Laboratory. We have selected a few — in our judgment the most typical — experi- ments in each subject, and our aim has been to enable the student to make use of his practical work to obtain a clearer and more real insight into the principles of the subjects. With but few exceptions, the experiments selected are of an elementary character ; they include those which have formed for the past three years our course of practical physics for the students preparing for the first part of the Natural Sciences Tripos ; to these we have now added some ex- periments on acoustics, on the measurement of wave-lengths, xii Preface. and on polarisation and colours. Most of the students have found it possible to acquire familiarity with the contents of such a course during a period of instruction lasting over two academical terms. The manner in which the subjects are divided requires perhaps a word of explanation. In conducting a class in- cluding a large number of students, it is essential that a teacher should know how many different students he can accommodate at once. This is evidently determined by the number of independent groups of apparatus which the Laboratory can furnish. It is, of course, not unusual for an instrument, such as a spectrometer, an optical bench, or Wheatstone bridge, to be capable of arrangement for working a considerable number of different experiments ; but this is evidentiy of no assistance when the simultaneous accommo- dation of a number of students is aimed at For practical teaching purposes, therefore, it is an obvious advantage to divide the subject with direct reference to the apparatus required for performing the different experiments. We have endeavoured to carry out this idea by dividing the chapters into what, for want of a more suitable name, we have called * sections,' which are numbered continuously throughout the book, and are indicated by black type headings. Each section requires a certain group of apparatus, and the teacher knows that that apparatus is not further available when he has assigned the section to a particular student. The different experiments for which the same apparatus can be employed are grouped together in the same section, and indicated by italic headings. The proof-sheets of the book have been in use during the past year, in the place of the original MS. books, m the following manner : — The sheets, divided into the sectioiu Preface, ^ above mentioned, have been pasted into MS. books, the re- maining pages being available for entering the results obtained by the students. The apparatus referred to in each book is grouped together on one of the several tables in one large room. The students are generally arranged in pairs, and be- fore each day's work the demonstrator in charge assigns to each pair of students one experiment — that is, one section of the book. A list shewing the names of the students and the experiment assigned to each is hung up in the Laboratory, so that each member of the class can know the section at which he is to work. He is then set before the necessary apparatus with the MS. book to assist him ; if he meets' with any difficulty it is explained by the demonstrator in charge The results are entered in the books in the form indicated for the several experiments. After the class is over the books are collected and the entries examined by the demonstrators. If the results and working are correct a new section is assigned to the student for the next time ; if they are not so, a note of the fact is made in the class list, and the student's attention called to it, and, if necessary, he repeats the experiment The list of sections assigned to the different students is now completed early in the day before that on which the class meets, and it is hoped that the publication of the description of the experiment will enable the student to make himself acquainted beforehand with the details of his day's work. Adopting this plan, we have found that two demon- strators can efficiently manage two classes on the same day, one in the morning, the other in the afternoon, each con- taining from twenty-five to thirty students. The students have hitherto been usually grouped in pairs, in consequence of the want of space and apparatus. Although this plan XIV Preface. has some advantages, it is, we think, on the whole, undesir- able. We have given a form for entering results at the end of each section, as we have found it an extremely convenient, if not indispensable, arrangement in our own case. The numerical results appended as examples are taken, with very few exceptions, from the MS. books referred to above. They may be found useful, as indicating the d^ee of accuracy that is to be expected from the various experi- mental methods by which they are obtained. In compiling a book which is mainly the result of Labora- tory experience, the authors are indebted to friends and fellow- workers even to an extent beyond their own knowledge. We would gladly acknowledge a large number of valuable hints and suggestions. Many of the useful contrivances that facilitate the general success of a Lalxnatory in which a large class works, we owe to the Physical Laboratory of Berlin ; some of them we have described in the pages that follow. For a number of valuable suggestions and ideas we are especially indebted to the kindness of Lord Rayleigh, who has also in many other ways afforded us facilities for the development of the plans and methods of teaching explained above. Mr. J. H. Randell, of Pembroke College, and Mr. H. M. Elder, of Trinity College, have placed us under an obligation, which we are glad to acknowledge, by reading the proof-sheets while the work was passing through the press. Mr. Elder has also kindly assisted us by photograph* ing the verniers which are represented in the frontispiece. Cavendish Laboratory i December i, 1884. R. T. GLAZEBROOK. W. N. SHAW. CONTENTS. CHAPTER I. PHYSICAL MEASUREMENTS. fAQi Direct and indirect Method of Measurement . • • • i Indirect Measurements reducible to Determinations of Length and Mass . . . 4 Origin of the Similarity of Observations of Different Quantities . 7 CHAPTER II. UNITS OF MEASUREMENT. Method of expressing a Physical Quantity 9 Arbitrary and Absolute Units . • . . , • • 10 Absolute Units 13 Kondamental Units and Derived Units 17 Abeolnte Systems of Units 17 The CG.S. System 21 Arbitrary Units at present employed 22 Changes from one Absolute System of Units to another. Dimen- sional Equations 24 ConTersion of Quantities expressed in Arbitrary Units • • . 28 CHAPTER III. PHYSICAL ARITHMETIC Approximate Measurements « • 30 Errors and Corrections , j, • 31 Mean of Observations • • 32 Possible Accuracy of Measurement of different Quantities • . 35 Arithmetical Manipulation of Approximate Values . . • 36 Facilitation of Arithmetical Calculation by means of Tables. Interpolation. 40 a xvi Contents n Algebraical Approximation — ^Approximate Formulae — Introduc- tion of small Corrections 41 Application of Approximate Formulae to the Calculation of the Effect of Errors of Observation • . • • . • 44 Graphical Methods • . . ' , > . • • • 49 The Slide Rule • • • 51 CHAPTER IV. • MEASUREMENT OF THE MORE SIMPLE QUANTITIES. SBCTION Lbnoth Mbasurembnts 59 1. The Calipers 59 2. The Beam-Compass 63 3. The Screw-Guage 66 4. The Spherometer 68 5. The Reading Microscope — Measurement of a Base-Line . 73 A. The Kathetometer Microscope . . . . . • 77 6. The Kathetometer . . 78 Adjustments ..••••• 79 Method of Observation . • • • 83 Measurement of Areas ...... 85 7. Simpler Methods of measuring Areas of Plane Figures . . 85 Orthogonal Profection ...... 87 8. Determination of the Area of the Cross-section of a Cylin- drical Tube — Calibration of a Tube . . . . 89 Measurement of Volumes 92 9. Determination of Volumes by Weighing . . • . 92 10. Testing the Accuracy of the Graduation of a Burette . • 93 Measurement of Angles . . . . . . 94 Measurement of Solid Angles .... 94 Measurements of Time 96 11. Rating a Watch by means of a Seconds-Qock . • • 97 CHAPTER V. MEASUREMENT OF MASS AND DETERMINATION OF SPECIFIC GRAVITIES. 11. The Balance • • • 99 General Considerations 99 T7u Sensitiveness of a Balance ..... lOQ Contents. xvii PAGI 7%4 At^ustmeni of a Balance .... 103 PraetUal Details of Manipulaiion — Method of Osdllations 107 ta. Testing the Adjustments of a Balance .... 114 DeUrmination of the' Ratio of the Arms of a Balance and of the true Mass of a Body when the Arms of the Balance are unequal 116 Comparison of the Masses of the Scale Pans . • 1 17 14. Coirection of Weighings for the Buoyancy of the Air • .119 Densities and Specific Gravities — Definitions . 121 15. The Hydrostatic Balance 123 Determttuttion of the Specific Gravity of a Solid heavier than Water 123 Determinaiion of the Specific Gravity of a Solid lighter than Water 125 Determination of the Specific Gravity of a Liquid . 127 16. The Specific Gravity Bottle 128 Determination of the Specific Gravity of small Frag- ments of a Solid 128 Determination of the Specific Gravity of a Powder . 132 Determination of the Specific Gravity of a Liquid . 132 17. Nicholson's Hydrometer 133 Determination of the Specific Gravity of a Solid . 133 Determination of the Specific Gravity of a Liquid . 135 iS. Jolly's Balance 136 Determination of the Mass and Specific Gravity of a small Solid Body 137 DetermituUion of the Specific Gravity of a Liquid , 138 1^ The Common Hydrometer . . . • . • 139 Method of comparing the Densities of two Liquids by the Aid of the Kathetometer . • • . . 141 CHAPTER V*. MEASUREMENT OP VELOCITY AND ACCELERATION. B. Measurement of the Velocity of a Pendulum . • . 144 C Tracing the Curve described by a Falling Body and the Chaiscter of its Downward Acceleration. • . 148 1 1 xviH Contents. CHAPTER VL MECHANICS OF SOLIDS. aa The Pendulum . . . . . • .... 152 DetermituUion of the AcceUrcUion of Gravity by Pendulum Observations 152 Comparison of the Times of Vibration of two Pen- dulums — Method of Coincidences — Katet^s Pern' dulum ••••...• 155 21. Atwood's Madiine •••«•••• 160 D. The Fly-wheel ..••••.. 166 Moment of Inertia • 166 £. Pendulum of any shape 171 F. Ballistic Pendulum— Measurement of Moment of Momen- tum and of Momentum 174 Measurement of Times of Oscillation . • .180 Measurement of First Swing . . . . . 180 G. Funicular Polygoil— Graphic Method of Comparing Forces 182 Summary of thb General Theory of Elasticity . 185 22. Young's Modulus 187 Modulus of Torsion • 190 23. Maxwell's Vibration Needle 191 Observation of the Time of Vibration • • . 193 CakukUion of the Alteration of Moment of Inertia . 195 CHAPTER VII. MECHANICS OF LIQUIDS AND GASES. Measurement of Fluid Pressure . . . • • • 197 24. The Mercury Barometer 198 Setting and reading the Barometer , . . . 199 Correction of the Observed Height for Tempera- ture, ^c, 200 25. The Aneroid Barometer .•••••• 202 Measurement of Heights • • • • • 203 26. The Volumenometer 205 Verification of Boy l^s Law 206 Determination of the Density of a Solid . • . 209 H. Capillarity 210 Measurement of the Surface Tension of a Liquid . 210 L Worthington's Capillary Multiplier 215 Contents, lix CHAPTER VIII. ACOUSTICS. SBCnON PAGB De6mtioiis, &c • . 218 27. Comparison of the Pitch of Tuning-forks— Adjustment of two Forks to Unison . 219 28. The Siren 222 39. Determination of the Velocity of Sound in Air by Measure- ment of the Length of a Resonance Tube corresponding to a given Fork 226 ja Verification of the Laws of A^bration of Strings— Determina- tion of the Absolute Pitch of a Note by the Monochord 229 31 Determination of the Wave-Length of a high Note in Air hf means of a Sensitiye Flame . . . • • 234 CHAPTER IX. THSRMOMBTRY AND EXPANSION. Measurement of Temperature 337 3s. CoDstmctioQ of a Water Thermometer • . « • 244 3^ Thermometer Testing .•.«.. 34. Determination of the Boiling Point of a Liquid • 35. Determination of the Fusing Point of a Solid K. Eaects of Dissolved SalU on the Freezing Point COBFPICIINTS OF EXPANSION 36. Determination of the Coefficient of Linear Expansion of a Rod 37. The Weight Thermometer 38. The Constant Volume Air Thermometer • 247 250 251 253 255 257 259 265 Lk The Constant Pressure Air Thermometer • , • • 268 CHAPTER X CALORIMETR Y. 39. The Method of Mixture ..•••• 272 Determination of the Specific Heat of a Solid . • 272 Determination of the Specific Heat of a Liquid . 278 Determination of th^ Latent Heat of Water • .279 Determination of the Latent Heat of Steam , . 281 XX Contents. tBCnON PACK 4a The Method of Cooling . . • • • . • 285 M. Method of Cooling— Graphic Method of Calciilatioii . 288 N. DetenninatioD of the Mechanical Equivalent of Heat • . . 290 CHAPTER XL TENSION OF VAPOUR AND HYGROMBTRY. 41. Dalton's Experiment on the Pressure of Mixed Gases and Vapours 294 Hygromstry 299 42. The Chemical Method of determining the Density of Aqueous Vapour in the Air 301 43. Dines's Hygrometer — The Wet and Dry Bulb Thermometers 306 44. R^;nault*s Hygrometer 309 CHAPTER XII. PHOTOMETRY. 45. Bunsen's Photometer . . . • • • 312 46. Rumford's Photometer ....... 316 CHAPTER XHI. REFLEXION AND REFRACTION— MIRRORS AND LENSES. 47. Verification of the Law of Reflexion of Light . .318 48. The Sextant 32a O. Refraction of Light through a Plat^ and through a Prism . 328 VeriJUation of the Law of Refraction . . . 329 Caustic Curve by Refraction ... . . . 330 To find the Refractive Index 331 Verificaticn of the Law of Reflexion . • . 332 Refraction through a Prism 333 Optical Mbasurbmbnts 335 49. Measurement of the Focal Length of a Concave Mirror . • 337 5a Measurement of the Radius of Curvature of a Reflecting Surface by Reflexion 339 Measurement of Focal Lengths of Lenses . . . . 343 51. Measurement of the Focal Length of a Convex Lens (First Method) ......... 343 $9. Measurement of the Focal Length of a Convex Lens (Second Method) . 344 Contents. xxi mcnoM PACT 53. Measurement of the Focal Length of a Convex Lens (Third Method) 345 54. Measurement of the Focal Length of a Concave Lens . > 35^ P. Focal Lengths— Additional Methods of Measurement . 352 55. Focal Lines 354 Q. Focal Lines formed by a Prism ..... 356 Magnifying Powers of Optical Instruments . • . . 358 56. Measurement of the Magnifying Power of a Telescope (First Method) 359 57. Measurement of the Magnifying Power of a Telescope (Second Method) 361 5& Measurement of the Magnifying Power of a Lens or M a Microscope . 363 59. The Testing of Plane Surfaces . . • • • • 3^7 CHAPTER XIV. SPECTRA, REFRACTIVE INDICES AND WAVE-LENGTHS. Pure Spectra ...••.... 375 6a The Spectroscope .•••.*.. 377 Mapping a Spectrum ... • . . 377 Comparison of Spectra . . • . . . 381 Reiractiye Indices 382 61. Measurement of the Index of Refraction of a Plate by means of a Microscope 383 6s. The Spectrometer 385 Tke Adjustment of a Spectrometer . • . . 386 Measurements rtnth the Spectrometer .... 388 (1) Verification of the Law of Reflexion . . . 388 (2) Afeasurement of the Angle of a Prism . . 388 (3) Measurement of the Refractive Index of a Prism i^First Method) 389 Measurement of the Refractive Index of a Prism {Second Method) ...... 393 (4) Measurement of the Wave-Length of Light by means of a Diffraction Grating . . . . 395 63. The Optical Bench 398 Measurement of the Wave- Length of Light by means of FresnePs Bi»prism 399 Diffraction Experiments 404 xxli Contents, CHAPTER XV. POLARISED LIGHT, SECTION PAOl On the Determination of the Position of the Plane of Polarisation .•••...•• 405 64. The Bi-quartz 407 65. Shadow Polarimeteis * • • . • • • 41a CHAPTER XVL COLOUR VISION. 66. The Coloar Top 417 67. The Spectro- Photometer • • • • t • • 421 68. The Colour Box ........ 425 R. Colour Photometry . • . ... • • • 427 CHAPTER XVII. MAGNETISM. Properties of Magnets • • 433 Definitions 434 Magnetic Potential .•••.... 439 Forces on a Magnet in a Uniform Field . . • • 441 Magnetic Moment of a Magnet . • . . • 442 Potential due to a Solenoidal Magnet . . . . • 444 Force due to a Solenoidal Magnet . . . . • 445 Action of one Solenoidal Magnet on another • . . 447 Measurement of Magnetic Force 450 Magnetic Induction . 452 6q. Experiments with Magnets ..••.. 453 (a) Magnetisaiixnt of a Steel Bar • . • . 453 (b) Comparison of the Magnetic Moment of the same Magnet after different Methods of Treat ment^ or of two different Magnets . . . . 456 (c) Comparison of the Strengths of different Magnetic Fields of approximately Uniform Intensity , 459 (d) Measurement of the Magnetic Moment ^ a Magnet and of the Strength of the Field in which it hangs 459 (#) Determination of the Magnetic Moment of a Magnet oj any shape 46 1 Contents, xxiii sBcncm rAGB (/) DeUnninaiion of the Direction of the Marthas HoriaonitU Force 461 (^) Experiments on two Magnets— Comparison of Magnetic Moments 464 S, Comparison of Grayitatioiial and Magnetic Forces . . 467 T. Gaoss* Verification of the Law of Magnetic Force . . 470 U. Magnetic Induction due to the Earth ..... 474 7a Exploration of the Magnetic Field due to a given Magnetic Distribution 476 V. Magnetic Induction due to Iron 479 Deierminaium of the Magnetic Moment of a Soft Iron Rod 481 Determination of the Magnetic Susceptibility of a Soft Iron Rod 482 Magnetic Cycles— Hysterisis . • • • . 484 CHAPTER XVIII. ELECTRICITY— DEFINITIONS AND EXPLANATIONS OF ELECTRICAL TERMS. Conductors and Non-conductors • • . . • . 488 Resultant Electrical Force 488 Electromotiye Force 489 Electrical Potential. •••••.. 489 Current of Electricity 492 CG.S. Absolute Unit of Current • • • • 494 Sine and Tangent Galvanometers • • • . . 496 CHAPTER XIX. EXPERIMENTS IN THE FUNDAMENTAL PROPERTIES OF ELECTRIC CURRENTS—MEASUREMENT OF ELECTRIC CURRENT AND ELECTROMOTIVE FORCE. 71. Absolute Measure of the Current in a Wire • . 497 Galvanombtkrs . 501 Galvanometer Constant ...••.. 503 Reduction Factor of a Galvanometer ..... 507 Sensitiveness of a Galvanometer ..... 508 The Adjustment of a Reflecting Galvanometer . . . 510 72. Determination ci the Reduction Factor of a Galvano- meter . • 5" xxtv Contents, tBCTION VAC« Electrolysb 51a Definition of Electro-chemical Equivalent • • 5>2 73. Faraday's Law — Comparison of Electro^emical Equiva- lents . . . • 517 74. Joule's Law — Measurement of Electromotive Force . . 522 CHAPTER XX. ohm's law— comparison of electrical resistances AND electromotive FORCES. Definition of Electrical Resistance • • . • • 527 Series and Multiple Arc •••}.•• 528 Shunts 530 Absolute Unit of Resistance 531 Standards of Resistance 532 Resistance Boxes • . 533 Relation between the Resistance and Dimensions of a Wire of given Material 534 Specific Resistance 535 75. Comparison of Electrical Resistances • • • • • 536 76. Comparison of Electromotive Forces • • • • 541 77. Wheatstone's Bridge • 543 Measuremeni of Resistance 549 Measurement efa Galvanometer Resistance — Hufm- son^s Method 551 Measurement of a Battery Resistance -Moneys Method .553 78. The British Association Wire Bridge 557 Measurement of Electrical Resistance . • • 557 79. Carey Foster's Method of Comparing Resbtances • • 561 Calibration of a Bridge^ Wire .... 566 8a Poggendorff 's Method for the Comparison of Electromotive Forces— Latimer Clark's Potentiometer . ... 567 W. The Clark Cell 572 To Set up a Clarh Cell 572 The Clark Cell as a Standard of Electromotive Force 577 Measuring a Current .••.•. 578 K. The Silver Voltameter 579 Method of making a MeasuremetU • , . 580 • Contents. xxv CHAPTER XXI. GALVANOMETRIC MEASUREMENT OF A QUANTITY OF ELECTRICITY. Theory of the Method 582 Relatiim beitueen the QuoMiity of EUctricUy which fassis through a Galvanometer and the iniHal Angular Velocity produced in the Needle . . 582 Work done in turning the Magnetic Needle through a given Angle 59^') Electrical Accumulators or Condensers • • . . 586 Definition of the Capacity of a Condenser • • . 587 The Unit of Capacity 587 On the Form of Galvanometer suitable for the Comparison ofCapadtieft 588 81. Comparison of the Capacities of two Condensers . . . 589 (1) Approximate Method 589 (2) Null Method 592 82. Measurement in Absolute Measure of the Capacity of a Condenser 595 CHAPTER XXII. ELECTROBfAGNETIC INDUCTION. Propositions for determining the Electromagnetic Induc- tion 598 Y. Experimental Laws of Electromagnetic Induction . . 604 (1) The quantity of Electricity traversing the Secondary is directly proportioned to the Primary Current. 60$ (2) The quantity of Electricity traversing the Secondary is inversely proportional to the whole resistance of the Secondary . . . 608 (j) The quantity of Electricity traversing the Secondary depends on the mutual position of the trtfo Circuits 609 (4) Examination of the effect of the Medium near the Coils 610 (5) Induction due to the Motion of a Magnet . .611 L Comparison of a Coefficient of Mutual Induction and the Product of a Resistance and a Time .... 613 % xxvi Contents. SECTION PAGS r. Comparison of a Coefficient of Mutoal Induction with the Capacity of a Condenser and the Product of two Re- sbtances 6i6 A. . Comparison of two Coefficients of Mutual Induction . .618 e. The Earth Inductor 624 Determination of the Dip by the Earth Inductor . 626 Measurement 0/ the Magnetic Indtution at any point of a Magnetic Field by means of an Induced Current 626 INDEX 629 t E LNIVhRSlTY PRACTICAL PHYSICS. CHAPTER I. PHYSICAL MEASUREMENTS. The greater number of the physical experiments of the present day and the whole of those described in this book consist in, or involve, measurement in some form or other. Now a physical measurement — a measurement, that is to say, of a physical quantity — consists essentially in the comparison of the quantity to be measured with a unit quantity of the same kind. By comparison we mean here the determination of the number of times that the unit is contained in the quantity measured, and the number in question may be an integer or a fraction, or be composed of an integral part and a fractional part In one sense the unit quantity must remain from the nature of the case perfectly arbitrary, although by general agreement of scientific men the choice of the unit quantities may be determined in accordance with certain general prin- ciples which, once accepted for a series of units, establish cer- tain relations between the units thus chosen, so that they form members of a system known as an absolute system of units. For example, to measure energy we must take as our unit the energy of some body under certain conditions, but when we agree that it shall always be the energy of a body on which a unit force has acted through unit space, our choice has been exercised, and the unit of energy is no longer arbitrary, but B 3 Practical Physics, [Chai». L defined, as soon as the units of force and space are agreed upon ; we have thus substituted the right of selection of the general principle for the right of selection of the particular unit We see, then, that the number of physical units is at least as great as the number of physical quantities to be measured, and indeed under different circumstances several different units may be used for the measurement of the same quantity. The physical quantities may be suggested by or related to phenomena grouped under the different headings of Mechanics, Hydro-mechanics, Heat, Acoustics, Light, Electricity or Magnetbm, some being related to phenomena on the common ground of two or more such subjects. We must expect, therefore, to have to deal with a very large number of physical quantities and a correspond- ingly large number of units. The process of comparing a quantity with its unit — the measurement of the quantity— may be either direct or in- direct, although the direct method is available perhaps in one class of measurements only, namely, in that of length measurements. This, however, occurs so frequently in the different physical experiments, as scale readings for lengths and heights, circle readings for angles, scale readings ias galvanometer deflections, and so on, that it will be well to consider it carefully. The process consists in laying off standards against the length to be measured. The unit, or standard length, in this case is the distance under certain conditions of temperature between two marks on a bar kept in the Standards Office of the Board of Trade. This, of course, cannot be moved from place to place, but a portable bar may be obtained and com- pared with the standard, the difference between the two bemg expressed as a fraction of the standard. Then we may apply the portable bar to the length to be measured, deter- mining the number of times the length of the bar is contained in the given length, with due allowance for temperature, and Chap. L] Physical Measurements, 3 thus express the ^ven length in terms of the standard by means of successive direct applications of the fundamental method of measurement Such a bar is known as a scale or rule. In case the given length does not contain the length of the bar an exact number of times, we must be able to determine the excess as a fraction of the length of the bar ; for this purpose the length of the bar is divided by transverse marks into a number of equal parts — say lo-* each of these again into lo equal parts, and perhaps each of these stiU further into lo equal parts. Each of these smallest parts will then be -^^ of the bar, and we can thus determine the number of tenths, hundredths, and thousandths of the bar contained in the excess. But the end of the length to be measured may still lie between two consecutive thou- sandths, and we may wish to carry the comparison to a still greater accuracy, alUiough the divisions may be now so small that we cannot further subdivide by marks. We must adopt some different plan of estimating the fraction of the thousandth. The one most usually employed is that of the * vernier.' An account of this method of increasing the acciu-acy of length measurements is given in § i. This is, as already stated, the only instance usually oc- curring in practice of a direct comparison of a quantity with Its unit The method of determining the mass of a body by double weighing (see § 13), in which we determine the number of units and fractions of a unit of mass, which to- gether produce the same effect as was previously produced by the mass to be measured, approaches very nearly to a direct comparisoa And the strictly analogous method of substitution of units and fractions of a unit of electrical re- sistance, until their effect is equal to that previously produced by the resistance to be measured, may also be mentioned, as well as the measurement of time by the method of coinci- dences (§ 20). But in the great majority of cases the comparison is far from direct The usual method of proceeding is as follows i-^- B3 4 Practical Physics. (Chap. 1. An experiment is made the result of which depends upon the relative magnitude of the quantity and its unit, and the nume- rical relation is then deduced by a train of reasoning which may, indeed, be strictly or only approximately accurate. In the measurement, for instance, of a resistance by Wheatstone's Bridge, the method consists in arranging the unknown resbt- ance with three standard resistances so chosen that under cer- tain conditions no disturbance of a galvanometer b produced. We can then determine the resistance by reasoning based on Ohm's law and certain properties of electric currents. These indirect methods of comparison do not always afford perfectly satisfactory methods of measurement, though they are sometimes the only ones available. It is with these in- direct methods of comparing quantities with their units that we shall be mostly concerned in the experiments detailed in the present work. We may mention in passing that the consideration of the experimental basis of the reasoning on which the various methods depend forms a very valuable exercise for the student As an example, let us consider the determination of a quantity of heat by ^e method of mixture (§ 39). It is usual in the rougher experiments to assume (i) that the heat absorbed by water is proportional to the rise of temperature ; (2) that no heat is lost from the vessel or calorimeter ; (3) that in case two thermometers are used, their indications are identical for the same temperature. All these three points may be con- sidered with advantage by those who wbh to get clear ideas about the measurement of heat Let us now turn our attention to the actual process in which the measurement of the various physical quantities consists. A little consideration will show diat, whether the quantity be mechanical, optical, acoustical, magnetic or dectric, the process really and truly resolves itself into measuring certain lengths, or masses.' Some examples will * See articles by Clifford and Maxwell : ScientiJU AppartUus, Hand* hook to the Special Loan ColUction^ 1876, p. 55. Chap. I.] Physical Measurements, 5 make this sufficiently clear. Angles are measured by read- ings of length along certain arcs ; the ordinary measure- ment of time is the reading of an angle on a clock face or the space described by a revolving drum ; force is measured by longitudinal extension of an elastic body or by weighing ; pressure by reading the height of a column of fluid sup- ported by it ; differences of temperature by the lengths of a thermometer scale passed over by a mercury thread ; heat by measuring a mass and a difference of temperature ; lu- minous intensity by the distances of certain screens and sources of light ; electric currents by the angular deflection of a galvanometer needle ; coefficients of electro-magnetic induction also by the angular throw of a galvanometer needle. Again, a consideration of the definitions of the various physical quantities leads in the same direction. Each physical quantity has been defined in some way for the purpose of its measurement, and the definition is insuffi- cient and practically useless unless it mdicates the basis upon which the measurement of the quantity depends. A diefinition of force, for instance, is for the physicist a mere arrangement of words unless it states that a force is mea- sured by the quantity of momentum it generates in the unit of time ; and in the same way, while it may be interest- ing to know that ' electrical resistance of a body is the oppo- sition it offers to the passage of an electric current,' yet we have not made much progress towards understanding the precise meaning intended to be conveyed by the words * a resistance of 10 ohms,' until we have acknowledged that the ratio of the electromotive force between two points of a con- ductor to the current passing between those points is a quan- tity which is constant for the same conductor in the same physical state, and is called and is the * resistance ' of the conductor ; and, further, this only conveys a definite mean- ing to our minds when we understand the bases of measure- ment suggested by the definitions of electromotive force and electric current Practical Physics. [Crap. I. When the quantity is once defined, we may possibly be able to choose a unit and make a direct comparison ; but such a method is very seldom, if ever, adopted, and the measurements really made in any experiment are often sug- gested by the definitions of the quantities measured The following table gives some instances of indirect methods of measurement suggested by the definitions of the quantities to be measured. The student may consult the descriptions of the actual processes of measurement detailed in subsequent chapters : — Name of quantity measoTMl Meatarement actually made Mechanics. Area . • • Length (§ i-6). Volume . . • • Length. Velocity . • • Length and time. Acceleration . • • Velocity and time. Force . . • • Mass and acceleration, or extensioii of spring. Work . • Force and length. Energy . • Work, or mass and velocity. Fluid pressure (In abso- lute units) . Force and area (§ 24-26). Coefficients of elasticity Stress and strain, ue, force, and length or angle (§§ 22, 23). Sound. Velocity Length and time (§ 29). Pitch • Time (§ 28). Hbat. Temperature . • Length (§ 32). Quantity of heat • Temperature and mass (§ 39). Conductivity • • Temperature, heat, length, and time. Light. Index of refraction . Angles (§ 62). Intensity length (5 45). Magnetism. Quantity of magnetism Force and length (J 69). Intensity of field . Force and quantity of masnetism (S69). ^ Magnetic moment • • Quantity of maimetism and Irncrth (S69). Chap. L] Physical Measurements. j NaoM of qoanticy measured MeMoraneiiti •cCiiall j made Electricitt. Electric current • • Quantity of magnetism, force» and length (I 71). Qoantity of Electricity « Current and time ({ 72). Electromotive force « Quantity of electricity and work Resistance . . Electric current and £. Bi. F. ({ 7^ ). Electro-chemical equivalent Mass and quantity of electria^ (§72). The quantities given in the second column of the table are often such as are not measured directly, but the basis of measiu:ement has, in each case, already been given higher up m the table. If the measurement of any quantity be reduced to its ultimate form it will be found to consist always in measurements of length or mass. ' The measurement of time by counting ' ticks ' may seem at first sight an exception to this statement, but further consideration will shew that it, also, depends ultimately upon length measurement As £ar as the apparatus for making the actual observations is concerned, many experiments, belonging to different subjects, often bear a striking similarity. The observing apparatus used in a determination of a coefficient of tor- sion, the earth's horizontal magnetic intensity, and a coefficient of electro-magnetic induction, are practically identical in each case, namely, a heavy swinging needle and a telescope and scale ; the difference between the experi- ments consists in the difference in the origin of the forces which set the moving needle in motion. Many similar in- stances might be quoted. Maxwell, in the work already referred to ('Scientific Apparatus,' p. 15), has laid down the grounds on which this analogy between the experiments in different branches of the subject is based. ' All the physical sciences relate to the passage of energy under its various forms from one body to another,' and, accordingly, ' The measurement of mass may frequently be resolved into that ci length. The method of double weighing, however, U a fundamental measurement ndgemris. 8 Practical Physics. [Chap. 1. all instruments, or arrangements of apparatus, possess the following functions : — ' I. The Source of energy. The energy involved in the phenomenon we are studying is not, of course, produced from nothing, but enters the apparatus at a particular place which we may call the Source. * 2. The channels or distributors of energy, which cany it to the places where it is required to do work. '3. The restraints which prevent it from doing work when it is not required. * 4. The reservoirs in which energy is stored up when it is not required. *' 5. Apparatus for allowing superfluous energy to escape. * 6. Regulators for equalising the rate at which work is done. ' 7. Indicators or movable pieces which are acted upon by the forces under investigation. ' 8. Fixed scales on which the position of the indicator is read ofl^' The various experiments differ in respect of the functions included under the first six headings, while those under the headings numbered 7 and 8 will be much the same for all instruments, and these are the parts with which the actual observations for measurement are made. In some experi* ments, as in optical measurements, the observations are simply those of length and angles, and we do not compare forces at all, the whole of the measurements being ultimately length measurements. In others we are concerned with forces either mechanical, hydrostatic, electric or magnetic, and an experiment consists in observations of the magni- tude of these forces under certain conditions ; while, again, the ultimate measurements will be measurements of length and of mass. In all these experiments, then, we find a foundation in the fundamental principles of the measure- ment of length and of the measurements of force and mass. The knowledge of the first involves an acquaintance with Crap. I.] Physical Measurements. 9 some of the elementary properties of space, and to under- stand the latter we must have some acquaintance with the properties of matter, the medium by which we are able to realise the existence of force and energy, and with the pro- perties of motion, since all energy is more or less connected with the motion of matter. We cannot, then, do better than urge those who intend making physical experiments to begin by obtaining a sound knowledge of those principles c^ dynamics, which are included in an elementary account of the science of matter and motioa The opportunity has been laid before them by one — to whom, indeed, many other debts of gratitude are owed by the authors of this work — who was well known as being foremost in scientific book-writing, as well as a great master of the subject For us it will be sufficient to refer to MaxwelFs work on ' Matter and Motion ' as the model of what an introduction to the study of physics should be. CHAPTER IL UNITS OF MEASUREMENT. Method of Expressing a Physical Quantity. In considering how to express the result of a physical experi* ment undertaken with a view to measurement, two cases essentially different in character present themselves. In the first the result which we wish to express is a concrete physical quantity^ and in the second it is merely the ratio of two physical quantities of the same kind, and is accordingly a number. It will be easier to fix our ideas on this point if we consider a particular example of each of these cases, instead of discussing the question in general terms. Con- sider, therefore, the difference in the expression of the result of two experiments, one to measure a quantity of heat and the second to measure a specific heat — the measurements 10 Practical Physics. [Chap. VL of a mass and a specific gravity might be contrasted in a perfectly similar manner — ^in the former the numerical value will be different for every different method employed to express quantities of heat ; while in the latter the result, being a pure number, will be the same whatever plan of measuring quantities of heat may have been adopted in the course of the experiment, provided only that we have adhered through- out to the same plan, when once adopted In the latter case, therefore, the number obtained is a complete expression of the result, while in the former the numerical value alone conveys no definite information. We can form no estimate of the magnitude of the quantity unless we know also the unit which has been employed. The complete expression, therefore, of a physical quantity as distinguished from a mere ratio consists of two parts : (i) the unit quantity employed, and (2) the numerical part expressing the number of times, whole or fractional, which the unit quantity is contained in the quantity measured. The unit is a concrtU quantity of the same kind as that in the expression of which it is used. If we represent a quantity by a symbol, that must likewise consist of two parts, one representing the numerical part and the other representing the concrete unit A general form for the complete expression of a quantity may therefore be taken to be q [q], where q represents the numerical part and [q] the concrete unit. For instance, in representing a certain length we may say it is 5 [feet], when the numerical part of the expression is 5 and the unit [foot]. The number q is called the numerical measure of the quantity for the unit [q]. Arbitrary and Absolute Units, The method of measuring a quantity, q [q], is thus resolved into two parts : (i) the selection of a suitable unit [q], and (2) the determination of q, the number of times which this unit is contained in the quantity to be measured. The lecond part is a matter for experimental determination, and Crap. IT.] Units of Measurement. 1 1 has been considered in the preceding chapter. We proceed to consider the first part more dosely. The selection of [q] is, and must be, entirely arbitrary — that is, at the discretion of the particular observer who is making the measurement It is, however, generally wished by an observer that his numerical results should be under- stood and capable of verification by others who have not the advantage of using his apparatus, and to secure this he must be able so to define the unit he selects that it can be repro- duced in other places and at other times, or compared with the units used by other observers. This tends to the general adoption on the part of scientific men of common standards of length, mass, and time, although agreement on this point is not quite so general as could be wished. There are, however, two well-recognised standards of length* : viz. (i) the British standard yard, which is the length at 62° F. between two marks on the gold plugs of a bronze bar in the Standards Office ; and (a) the standard metre as kept in the IVench Archives, which is equivalent to 39*37079 British inches. Any observer in measuring a length adopts the one or the other as he pleases. All graduated instru- ments for measuring lengths have been compared either directly or indirectly with one of these standards. If great accuracy in length measurement is required a direct com- parison must be obtained between the scale used and the standard. This can be done by sending the instrument to be used to the Standards Office of the Board of Trade. There are likewise two well-recognised standards of mass , viz. (i) the British standard pound, a certain mass of platinum kept in the Standards Office ; and (2) the kilogramme des Archives, a mass of platinum kept in the French Archives, originally selected as the mass of one thou- sandth part of a cubic metre of pure water at 4° C One > See Maxwell's Heat, chap. iv. The British Standards are now kept at the Standards Office at the Board of Trade, Westminster, in iccordance with the * Weights and Measures Act^ 1878. 13 Practical Physics. (Chap. IL or other of these standards, or a simple fraction or multiple of one of them, is generally selected as a unit in which to measure masses by any observer making mass measure- ments. The kilogramme and the pound were carefully com- pared by the late Professor W. H. Miller ; one pound is equivalent to '453593 kilogramme. With respect to the unit of time there is no such divergence, as the second is generally adopted as the unit of time for scientific measurement The second is gglog of the mean solar day, and is therefore easily reproducible as long as the mean solar day remains of its present length. These units of length, mass, and time are perfectly arbi- trary. We might in the same way, in order to measure any other physical quantity whatever, select arbitrarily a unit quantity of the same kind, and make use of it just as we select the standard pound as a unit of mass and use it Thus to measure a force we might select a unit of force, say the force of gravity upon a particular body at a particular place, and express forces in terms of it This is the gravitation method of measuring forces which is often adopted in practice. It is not quite so arbitrary as it might have been, for the body generally selected as being the body upon which, at Lat 45'', gravity exerts the unit force is either the standard pound or the standard gramme, whereas some other body quite unrelated to the mass standards might have been chosen. In this respect the gallon, as a unit of measurement of volume, is a better example of arbitrariness. It contains ten pounds of water at a certain temperature. We may mention here, as additional examples of arbitrary units, the degree as a unit of angular measurement, the thermometric degree as the unit of measurement of tem- perature, the calorie as a unit of quantity of heat, the standard atmosphere, or atmo, as a unit of measurement of fluid pressure, Snow Harris's unit jar for quantities of electricity, and the B.A. unit' of electrical resistance Chap. II.] Units of MeasuretMeni. 13 Absolute Units. The difficulty, however, of obtaining an arbitrary standard irhich is sufficiently permanent to be reproducible makes this arbitrary method not always applicable. A fair example of this is in the case of measurement of electro-motive force,' for which no generally accepted arbitrary standard has yet been found, although it has been sought for very diligently. There are also other reasons which tend to make phjrsicists select the units for a large number of quantities with a view to simplifying many of the numerical calculations in which the quantities occur, and thus the arbitrary choice of a unit for a particular quantity is directed by a principle of selection which makes it depend upon the units already selected for the measiurement of other quantities. We thus get systems of units, such that when a certain number of fundamental units are selected, the choice of the rest follows from fixed principles. Such a system is called an ' absolute ' system of units, and the units themselves are often called 'absolute,'' although the term does not strictly apply to the individual units. We have still to explain the principles upon which absolute systems are founded. Nearly all the quantitative physical laws express relations between the numerical measures of quantities, and the general form of relation is that the numerical measure of some quantity, Q, b proportional (either directly or inversely) to certain powers of the numerical measures of the quan- tities X, Y, z . . . If ^, or, ^, 5, . . . be the numerical measures of these quantities, then we may generalise the physical law, and express it algebraically thus : g is propor- tional to :r*,y, i^, . . ., or by the variation equation qocx^.y^,^ , . . . where a, /?, y may be either positive or negative, whole or frac- tional The following instances will make our meaning clear: » Since thii was wriUen, it has been shewn that the E.M.F. of a Latimer-aark's cell is very nearly ccmstant, and equal to i -434 volt at ic« C. rSee o. ?72.^ 14 Practical Physics. [Chap. n. (i.) The volumes of bodies of similar shape are propor- tional to the third power of their linear dimensions, or (2.) The rate of change of momentum is proportional to the impressed force, and takes place in the direction in which the force is impressed (Second Law of Motion), or foQtna, (3.) The pressure at any point of a heavy fluid is propor- tional to the depth of the point, the density of the fluid, and the intensity of gravity, or poQhpg. (4,) When work produces heat, the quantity of heat produced is directly proportional to the quantity of work expended (First Law of Thermo-dynamics), or hocw. (5.) The force acting upon a magnetic pole at the centre of a circular arc of. wire in which a current is flowing, is directly proportional to the strength of the pole, the length of the wire, and the strength of the current, and inversely proportional to the square of the radius of the circle, or /* Ic and so on for all the experimental physical laws. We may thus take the relation between the numerical measures — qoQxry^sf . . . to be the general form of the expression ot an experimental law relating to physical quantities. This may be written in the form q^kxr/t (i) when ^ is a 'constant' This equation, as we have already stated, expresses a Chap. II.] Units of Measurement 1$ relation between the numerical measures of the quantities involved, and hence if one of the units of measurement is changed, the numerical measure of the same actual quan- tity will be changed in the inverse ratio, and the value of k will be thereby changed We may always determine the numerical value of k if we can substitute actual numbers for f, jc, j^, i^ • . . in the equation (i). For example, the gaseous laws may be expressed in words thus: — 'The pressure of a given mass of gas is directly pro- portional to the temperature measured from —273** C., and inversely proportional to the volume,' or as a variation equation — . B We may determine k for i gramme of a given gas, say hydrogen, from the consideration that i gramme of hydro- gen, at a pressure of 760 mm. of mercury and at o® C, occu- pies I 1200 cc Substituting / = 760, tf = 273, z; = 1 1 200, we get . 760x11200 o 273 and hence Q /=3ii8o- • . . (2). Here/ has been expressed in terms of the length of an equivalent column of mercury ; and thus, if for v and we substitute in equation (2) the numerical measures of any volume and temperature respectively, we shall obtain the corresponding pressure of i gramme of hydrogen expressed in millimetres of mercury. This, however, is not the standard method of expressing l6 Practical Physics. [Chap. n. a pressuie ; its standard expression is the force per unit of area. If we adopt the standard method we must substitute for/ not 760^ but 76 x 13*6 x 981, this being the numl>er of units of force * in the weight of the above column of mercury of one square-centimetre section. We should then get for k a different value, viz. : — X. 1,014,000x11200 *=-2 — -12 =41500000, so that /= 41 500000- . . . (3), and now substituting any values for the temperature and volume, we have the corresponding pressure of i gramme of hydrogen expressed in units of force per square centimetre. Thus, in the general equation (i), the numerical value of k depends upon the units in which the related quantities are measured ; or, in other words, we may assign any value we please to k by properly selecting the units in which the related quantities are measured. It should be noticed that in the equation we only require to be able to select one of the units in order to make k what we please ; thus x^y^ z^ . . . may be beyond our control, yet if we may give q any numeriod value we wish, by selecting its unit, Uien k may be made to assume any value required. It need hardly be mentioned that it would be a very great convenience if k were made equal to unity. This can be done if we choose the proper unit in which to measure q. Now, it very frequently happens that there is no other countervailing reason for selecting a different unit in which to measure Q, and our power of arbitrary selection of a unit for Q is thus exercised, not by selecting a particular quantity of the same kind as q as unit, ' The units of force here used are dynes or C.G.S. units of fofc«* Chap. II.] Units of Measurement. 17 and holding to it however other quantities may be mea- sured, but by agreeing that the choice of a unit for q shall be determined by the previous selections of units for X, T, z, . . . together with the consideration that the quantity k shall be equal to unity. Fundamental Units and Derived Units. It is found that this principle, when fully carried out, leaves us free to choose arbitrarily three units, which are therefore called fundamental units, and that most of the other units employed in physical measurement can be defined with reference to the fundamental units by the consider- ation that the factor k in the equations connecting them shall be equal to unity. Units obtained in this way are called derived units, and all the derived imits belong to an absolute system based on the three fundamental units. Absolute Systems of Units. Any three units (of which no one is derivable from the other two) may be selected as fundamental umts. In those systems, however, at present in use, the units of length, mass, and time have been set aside as arbitrary fundamental units, and the various systems of absolute imits differ only in regard to the particular units selected for the measure- ment of length, mass, and time. In the absolute system adopted by the British Asslodation, the fundamental units selected are the centimetre, the gramme, and the second re- spectively, and the system is, for this reason, known as the C.G.S. system. For magnetic surve3ring the British Government uses an absolute system based on the foot, grain, and second ; and scientific men on the Continent frequently use a system based on the millimetre, milligramme, and second, as fun- damental units. An attempt was also made, with partial success, to introduce into England a system of absolute units, based upon the foot, pound, and second as funda- mental units. c IS Practical Physict. [CHAP. IL ' 5 t t § ^ •s. i 3 3 3 V h k 1 h i i .l ■ Pi 1 \ I, ° Ssu d i i S^ 1 1 ~fl '1 ^li If.; n* 1^ 1 •5 1^ 15. SI III ■' lit hi 11 . «. *. ; ; ^ , $s 1' r " S 5 ' ^ ^ ^ 1 % s d .;, 1 Chap. IL] 19 i i 2 5 "^ 1 i ; 1 i i 1 i 1 'S 1' 11 1 rt* l-ifs i m Pll l^i S!i ^ 4 3- "iK » -J » i g- «, ■v ■ ■ 3 1 ii ■4 4. ft i„ ■si fri 111 Practical Physics. s i ! s III i 1 !i. sK 1 1 I ill p 111 ^|ii f It * * 1 •ilu 1 A i l! Ill i 11. Chaf. II.] Units of Measurement. 31 The C.G.S. System. The table, p^ 18, shows the method of derivation of such absolute units on the C.G.S. system as we shall have occasion to make use of in this book. The first column contains the denominations of the quantities measiured ; the second contains the verbal expression of the physical law on which the derivation is based, while the third gives the expression of the law as a variation equation ; the fourth and fifth columns give the definition of the CG.S. unit obtained and the name assigned to it respectively, while the last gives the dimensional equatioa This will be explained later (p. 24). The equations given in the third column are reduced to ordinary equalities by the adoption of the unit defined in the next colunm, or gmT another unit belonging to an absolute system based on the same principles. Some physical laws express relations between quantities whose units have already been provided for on the absolute system, and hence we cannot reduce the variation equations to ordinary equalities. This is the case with the formula for the gaseous laws already mentioned (p 15). A complete system of units has thus been formed on the CG^ absolute system, many of which are now in practical use. Some of the electrical units are, however, proved to be not of a suitable magnitude for the electrical measurements most firequently occurring. For this reason practical units have been adopted which are not identical with the C.G.S. units given in the table (p. 20), but are immediately derived firom them by multiplication by some power of 10. The names of the units in use, and the £u:tors of derivation firom the corresponding CG.S. units are given in the following table : — 33 Practical Physics. [Chap. n. Tablb of Practical Units for Elbctrical Mrasurbmbnt RELATED TO THB C.G.S. ElBCTRO-MAGNBTIC SYSTBM. Quantity Unit Equivalent m CG.S. onits Electric current Ampte 10 -» Electromotive force Vc^t I0« ResistaDce .. . Ohm I0» Capacity . Farad io-» Rate of working Watt I0» Quantity of Electricity . Coulomb 10 -> To shorten the notation when a very small fraction or a very laige multiple of a unit occurs, the prefixes micro- and megO' have been introduced to represent respectively divi- sion and multiplication by lo^ Thus: — A mega-dyne = lo* dynes. Amkro-farad lo' farad. Arbitrary Units at present employed. For many of the quantities referred to in the table (p. i8) no arbitrary unit has ever been used. Velocity, for instance, has always been meastured by the space passed over in a unit of time. And for many of them the physical law given in the second column b practically the definition of the quantity ; for instance, in the case of resistance. Ohm's law is the only definition that can be given of resistance as a measurable quantity. For the measurement of some of these quantities, how- ever, arbitrary units have been used, especially for quan- tities which have long been measured in an ordinary way as volumes, forces, &c Arbitrary units are still in use for the measurement of temperature and quantities of heat; also for light intensity, and some other magnitudes. We have collected in the following table some of the arbitrary imits employed, and given the results of experi- mental determinations of their equivalents in the absolute Chap. IL] Units of Measurement. 23 units for the measurement of the same quantity when sud exist: — Tabls of Arbitrary Units. Quantity Angle Force Work Temperature Quantity oi heat Intensity of light Electrical re- sistance ArUtnry imit «inpIoye<l Eqaivalent m absolute units (lis part of two right angles) Kadian (unit of circular measure) Pound weight Gramme weight Foot-pound Kilogramme-metre Degree Centigrade, corre- sponding to j^ of the expansion ot mercury in |;las8 between the fireezmg and boiling pdnts; degree Fahren- neity conesponding to ^ of the same quantity Amount of heat required to raise the temperature df unit mass of water one degree Standard candle. Sperm candles of nz to the pound, each burning 120 grains an hour TSt Paris Conference stan* daid. The light emitted by I sq. cm. of platinum at its melting point The B.A. unit (originally intended to represent the ohm) The <ohm' adopted by the Board of Trade. The resistance at o^ C. of a column of mercury 106 '3 cm. long, <» uniform cross-section, 14*4521 grms. in mass. 32*2 ponndals (British absolute units) 981 dynes 32*2 foot*poundals 9 '81 X 10^ ergs The gramme - centi- grade unit is equi- valent to 4*214x10' ergs *9866 true ohm ' > Cavendish Laboratory determinations. 24 Practical Physics. [Chap. IL Changes from one Absolute System of Units to another. Dimensional equations. We have already pointed out that there are more than one absolute system of units in use by physicists. They are deduced in accordance with the same principles, but are based on different values assigned to the fundamental units. It becomes, therefore, of importance to determine the factor by which a quantity measured in terms of a unit be- longing to one system must be multiplied, in order to express it in terms of the unit belonging to another system. Since the systems are absolute systems, certain variation equations become actual equalities ; and since the two systems adopt the same principles, the corresponding equations will have the constant k equal to unity for each system. Thus, if we take the equation (i) (p. 14) as a type of one of these equa- tions, we have the relation between the numerical measures q^osry^iS' holding simultaneously for both systems. Or, if ^, x^y^ g, be the numerical measures of any quan- tities on the one absolute system ; /, ^', y, «', the numerical measures of the same actual quantities on the other system, ^^^ q = j»-y r . . , . (i) and ^ = ^-y 5''^ • . . . (2). Now, following the usual notation, let [q], [x], [y], [z] be the concrete units for the measurement of the quantities on the former, which we will call the old, system, [q'], [x'], [y'], [z'] the concrete units for their measurement on the new system. Then, since we are measuring the same actual quantities^ x] s xf [x'] X y % Z] S «' \7l (3). * The 83rmbol s is used to denote absolute identity, as distinguished from numerical equality. Chap. IL] Units of Measurement, 25 In these we may see clearly the expression of the well-known law, that if the unit in which a quantity is measured be changed, the ratio of the numerical measures of the same quantity for the two units is the inverse ratio of the units. From equations (i) and (2) we get J=(J)'«0'(p): and substituting from (3). ^-([ii)-(^)'(^)' 2.= Thus, if ^, 17, { be the ratio of the new units [x^], [y^, \zr\ to the old units [xj \y\ [z] respectively, then the ratio p of the new unit \cf\ to the old unit [q] is equal to t^fl^y and the ratio of die new numerical measiure to the old is the reciprocal of this. Thus P^e-rf^ (4). The equation (4), which egresses the relation between the ratios in which the units are changed, is of the same form as (i), the original expression of the physical law. So that whenever we have a physical law thus expressed, we get at once a relation between the ratios in which the units are changed. We may, to avoid multiplying notations, write it, if we please, in the following form : — w^ere now [q], [x], [y], [z] no longer stand for concrete unitSy hut for the ratios in which the concrete units are flanged. It should be unnecessary to call attention to this, as it is, of course, impossible even to imagine the multiplication of one concrete quantity by another, but the constant use of the identical form may sometimes lead the student to infer that the actual multiplication or division of concrete quantities 26 Practical Phystcs. [Chap. II. takes place. If we quite clearly understand that the sen- tence has no meaning except as an abbreviation, we may express equation (5) in words by saying that the unit of Q is the product of the a power of the unit of x, the j3 power of the unit of Y, and the y power of the unit of z ; but if there is the least danger of our being taken at our word in express- ing ourselves thus, it would be better to say that the ratio in which the unit of Q is changed when the units of x, y, z are changed in the ratios of [x] : i, [y] : i and [z] : i re- spectively, is equal to the product of the a power of [x], the P power of [y], and the y power of [z]. We thus see that if [x], [y], [z] be the ratios of the new units to the old, then equation (5) gives the ratio of the new unit of Q to the old, and the reciprocal is the ratio of the new numerical measure to the old numerical measure. We may express this concisely, thus : — If in the equa- tion (5) we substitute for [x], [y], [z] the new units in terms of the old, the result is the factor by which the old unit of Q must be multiphed to give the new unit ; if, on the other hand, we substitute for [x], [y], [z] the old units in terms of the new, then the result is the factor by which the old numerical measure must be multiplied to give the new numerical measure. If the units [x], \y\ [z] be derived units, analogous equations may be obtained, connecting the ratios in which they are changed with those in which the fundamental units are changed, and thus the ratio in which [q] is changed can be ultimately expressed in terms of the ratios in which the fundamental units are changed. We thus obtain for every derived unit [q]=[l]-[m]'[t]' . . .(6). [l], [if], [t] representing the ratios in which the funda- mental units ik length, mass, and time, respectively, are 'hanged. The equation (6) is called the dimensional equation iot Chap. II.) Units of Measurement. 27 [q\ and the indices a, j3, y are called the dimensions of Q with respect to length, mass, and time respectively. The dimensional equation for any derived unit may thus be deduced from the physical laws by which the unit is defined, namely, those whose expressions are converted from variation equations to equalities by the selection of the unit. We may thus obtain the dimensional equations which are given in the last column of the table (p. 18). We give here one or two examples. (i) To find the Dimensional Equation for Velocity. Physical law s ^vty or Hence w = s r L i— i T = [L][T]-1. (2) To find the Difnensional Equation for Force. Physical law /=s ma. Hence W = [M] W ; but • • M] [L] [T]-». (3) ^ fi^ ^^ Dimensional Equation for Strength of Magnetic Pole. Physical law Hence /. M« = /«/ 28 Practical Physics.- |Chap. il But f = [ii][l][tJ-^ or M = [M]» [L]l [t]-'. Whenthe dimensional equations for the different units have been obtained, the calculation of the factor for con- version is a very simple matter, following the law given oo p. 26. We may recapitulate the law here. To find the Factor by which to multiply the Numerical Measure of a Quantity to convert it from the old System of Units to the new, substitute for [l] [mJ and [t] in the Ditnen- sional Equation the old Units of Lengthy Mass^ and Time respectively^ expressed in terms of the new. We may shew this by an example. To find the Factor for converting the Strength of a Mag- netic Pole from C.G.S to Foot-grain-second Units--^ I C.G.S. unit of magnetic pole = I X [M]i [l]* [t]-1 = I X [gm.]* [cm.]t [sec.]-i = I X [15-4 gr.]* [0-0328 ft]* [sec.]-i = I X (15-4)* (o-o328)t [gr.]4 [ft)t [sec.]-» = '0233 foot-grain-second unit That is, a pole whose strength is 5 in C.G.S. units has a strength of •1165 foot-grain-second units. Conversion of Quantities expressed in Arbitrary Units, We have shewn above how to change from one system of units to another when both systems are absolute and based on the same laws. If a quantity is expressed in Chap. II.] Units of Measurement. 29 arbitraiy units, it must first be expressed in a unit belonging to some absolute system, and then the conversion factor can be calculated as above. For example : — To express 15 Foot-pounds in Ergs. The foot-pound is not an absolute unit We must first obtain the amount of work expressed in absolute units. Now, since ^=32*2 in British absolute units, i foot-pound ^ 32*2 foot-poundals (British absolute units). •% 15 foot-pounds = 15 X 32*2 foot-poundals. We can now convert from foot-pouncfals to ergs. The dimensional equation is Since Substituting we get [w] = [m][lP[t]-«. I foot = 30*5 cm. I lb. = 4S4 gm. [m]=454, W=3o-S [w] = 454X(30-5)*. Hence IS foot-pounds = IS X 32-2 X 454 X (30-5)* ergs. = 2 •04x10* ergs. Sometimes neither of the units belongs strictly to an absolute system, although a change of the fundamental units alters the unit in question. For example : — To find the Mechanical Equivalent of Heat in C.G.S. Centigrade Units^ knowing that its Value for a Pound Fahrenheit Unit of Heat is 772 Foot-pounds. The mechanical equivalent of heat is the amount of work equivalent to one unit of heat For the CG.S. Centi- grade unit of heat, it is, therefore, 2x ^ X772 foot-pounds. 5 454 30 Practical Physics. [Chap. IIL This amount of heat is equivalent to ^x— X 772 xi'36xio' ergs, 5 454 or the mechanical equivalent of one C.G.S. Centigrade unit of heat = 4*i4X 10' ergs. If the agreement between scientific men as to the selection of fundamental units had been universal, a great deal of arithmetical calculation which is now necessary would have been avoided. There is some hope that in future one uniform system may be adopted, but even then it will be necessary for the student to be familiar with the methods of changing from one system to another in order to be able to avail himself of the results already published To form a basis of calculation, tables showing the equiva- lents of the different fundamental units for the measure* ment of the same quantity are necessary. Want of space prevents our giving them here ; we refer instead to Nos. 9-12 of the tables by Mr. S. Lupton (Macmillan & Co.). We take this opportunity of mentioning that we shall refer to the same work ^ whenever we have occasion to notice the necessity for a table of constants for use in the experiments described CHAPTER III. PHYSICAL ARITHMETIC Approximate Measurements. One of the first lessons which is learned by an experimenter making measurements on scientific methods is that the number obtained as a result is not a perfectly exact expres- sion of the quantity measured, but represents it only withic * Numerical Tables and CamUmU m Elementary Science^ faj & Luptoa CHAP, m.] Physical Arithmetic. 31 certain limits of error. If the distance between two towns be given as fifteen miles, we do not understand that the distance has been measured and found to be exactly fifteen miles, without any yards, feet, inches, or ftactions of an inch, but that the distance is nearer to fifteen miles than it is to sbcteen or fourteen. If we wished to state the distance more accurately we should have to begin by defining two points, one in each town — marks, for instance, on the door* steps of the respective parish churches — between which the distance had been taken, and we should also have to specify tiie route taken, and so on. To determine the distance with the greatest possible accuracy would be to go through the laborious process of measuring a base b'ne, a rough idea of which is given in § 5. We might then, perhaps, obtain the distance to the nearest inch and still be uncertain whether there should not be a fraction of an inch more or less, and if so, what fraction it should be. If the number is expressed in the decimal notation, the increase in the accuracy of measurement is shewn by filling up more decimal places. Thus, if we set down the mechanical equivalent of heat at 4*2 x 10^ ergs, it is not because the figures in the decimal places beyond the 3 are all zero, but because we do not know what their values really are, or it may be^ for the purpose for which we are using the value, it is immaterial what they are. It is known, as a matter of &ct, that a more accurate value is 4*214 x 10^, but at present no one has been able to determine what figure should be put in the decimal place after the second 4. Errors and Corrections, The determination of an additional figure in a number representing the magnitude of a physical quantity generally involves a very great increase in llie care and labour which must be bestowed on the determination. To obtain some idea of the reason for this, let us take, as an example, the case of determining the mass of a body of about 100 32 Practical Physics, [Chap. IIL grammes. By an ordinary commercial balance the mass d a body can be easily and rapidly determined to i gramme, say 103 grammes. With a better arranged balance we may shew that 103*25 is a more accurate representation of the mass. We may then use a very sensitive chemical balance which shews a difference of mass of o'l mgm., but which requires a good deal of time and care in its use, and get a value 103*2537 grammes as the mass. But, if now we make another similar determination with another balance, or even with the same balance, at a different time, we may find the result is not the same, but, say, 103 2546 grammes. We have thus, by the sensitive balance, carried the measurement two decimal places further, but have got from two observations two different results, and have^ there- fore, to decide whether either of these represents the mass of the body, and, if so, which. Experience has shewn that some, at any rate, of the difference may be due to the balance not being in adjustment, and another part to the fact that the body is weighed in air and not in vacuo. The observed weighings may contain errors due to these causes. The effects of these causes on the weighings can be cal- culated when the ratio of the lengths of the arms and other facts about the balance have been determined, and when the state of the air as to pressure, temperature, and moisture is known (see §§13, 14). We may thus, by a series of auxiliary observations, determine a correction to the observed weighing correspond- ing to each known possible error. When the observations are thus corrected they will probably be very much closer. Suppose them to be 103*2543 and 103*2542. Mean of Observations. When all precautions have been taken, and all known errors corrected, there may still be some difference between different observations which can only arise from causes beyond the knowledge and control of the observer. We Chap. III.] Physical Arithmetic. 33 most, therefore, distinguish between errors due to known causes, which can be allowed for as corrections, or elimi- nated by repeating the observations under different con- ditions, and errors due to unknown causes, which are called 'accidental ' errors. Thus, in the instance quoted, we know of no reason for taking 103*2543 as the mass of the body in preference to 103*2542. It is usual in such cases to take the arithmetic mean of the two observations, i.e. the number obtained by adding the two values together, and dividing by 2, as the nearest approximation to the true value. Similarly if any number, n^ of observations be taken, each one cf which has been corrected for constant errors^ and is, therefore, so far as the observer can tell, as worthy of confidence as any of the others, the arithmetic mean of the values is taken as that most nearly representing the true value of the quantity. Thus, if ^1, ^21 ^3 • • • • ^i» l>e the results of the n observations, the value of q is taken to b^ t m ■•?■'', •*. It is fair to suppose that, if we take a sufficient number of observations, some of them give results that are too large, others again results that are too small ; and thus, by taking the mean of the observations as the true value, we approach more nearly than we can be sure of domg by adopting any single one of the observations. We have already mentioned that allowance must be made by means of a suitable correction for each constant error, that is for each known error whose effect upon the result may be calculated or eliminated by some suitable arrangement It is, of course, possible that the observer may have overlooked some source of constant error which will affect the final result This must be very carefully guarded against, for taking the mean of a number of obser- D I " \ . I I 34 Practical Physics. [Chap. in. vations affords, in general, no assistance in the elimination of an error of that kind The difference between the mean value and one of the . observations is generally known technically as the 'error' of that observation. The theory of probabilities has been applied to the discussion of errors of observations ^ and it has been shewn that by taking the mean of n observations instead of a single observation, the so-called 'probable error' is reduced in the ratio of ifs/n. On this account alone it would be advisable to take several observations of each quantity measured in a physical experiment By doing so, moreover, we not only get a result which is probably more accurate, but we find out to what extent the observations differ from each other, and thus obtain valuable information as to the degree of accuracy of which the method of observation is capable. Thus we have, on p. 72, four observations of a length, viz. — 3*333 >n. 3332 n 3 334 ,. 3 334 >» Mean = 3333 2 „ Taking the mean we are justified in assuming that the true length is accurately represented by 3*333 to the third decimal place, and we see that the different observations differ only by two units at most in that place. In performing the arithmetic for finding the mean of a number of observations, it is only necessary to add those columns in which differences occur — the last column of the example given above. Performing the addition on the other columns would be simply multiplying by 4, by which number we should have subsequently to divide. An example will make this clear. * Sec Airy*s tnict on the Theory of Errors of Observations. Chap. ni.J Physical Arithmetic, 3S I^znd tlu mean of the following eight observations ;— - 56-231 56-275 56-243 56-255 56-256 56267 56*273 56*266 Adding (8 x 56*2 -|-)'466 Mean . . 56*2582 The figures introduced in the bracket would not appeai in ordinary working. The separate observations of a measurement should be made quite independently, as actual mistakes in reading are always to be regarded as being within the bounds of pos- sibility. Thus, for example, mistakes of a whole degree are sometimes made in reading a thermometer, and again in weighing, a begiimer is not unlikely to mis-count the weights. Mistakes of this kind, which are to be very care- fully distinguished from the * errors of observation,' would probably be detected by an independent repetition of the observatioa If there be good reason for thinking that an observation has been affected by an unknown error of this kind, the observation must be rejected altogether. Possible Accuracy of Measurement of different Quantities, The degree of accuracy to which measurements can be carried varies very much with different experiments. It is usual to estimate the limit of accuracy as a fractional part or percentage of the quantity measured. Thus by a good balance a weighing can be carried out to a tenth of a milligramme ; this, for a body weighing about 100 grammes, is as far as one part in a million, or *oooi per cent — an accuracy of very high order The measurement D 2 36 Practical Physics. [Chap. III. of a large angle by the spectrometer (§ 62) is likewise very accurate ; thus with a vernier reading to 20'', an angle of 45** can be read to one part in four thousand, or 0*025 per cent On the other hand, measurements of temperature cannot, without great care, be carried to a greater degree of accuracy than one part in a hundred, or I per cent, and sometimes do not reach that A length measurement often reaches about one part in ten thousand. For most of the experiments which are described in this work an accuracy of one part in a thousand is ample, indeed generally more than sufficient It is further to be remarked that, if several quantities have to be observed for one experiment, some of them nuy be capable of much more accurate determination than others. It is, as a general rule, useless to carry the accuracy of the former beyond the possible degree of accuracy of the latter. Thus, in determining specific heats, we make some weighings and measure some temperatures. It is useless to determine the weights to a greater degree of accuracy than one part in a thousand, as the accuracy of the result will not reach that limit in consequence of the inaccuracy of the temperature measurements. In some cases it is necessary that one measurement should be carried out more accurately than others in order that the errors in the result may be all of the same order. The reason for this will be seen on p 48. Arithmetical Manipulation of Approximate Values, In order to represent 1 quantity to the degree of accuracy of one part in a thousana, we require a number with four digits at most, exclusive of the zeros which serve to mark the position of the number in the decimal scale.* It frequently > It i^ now usual, i^h«B a very large number has to be expressed, to }^te^VfV^ the digits with a decimal point after the fiist, and indicate its po;>Ition in the scale by the power of 10, by which it must be mul- tiplied : thus, instead of 42140000 we write 4*214 x 10'. A corre- sponding notation is used for a very smaU decimal fraction : thus, mstead of '00000588 we write 588 x lO"*. Chap. III.] Physical Arithmetic 37 happens that some arithmetical process, employed to deduce the required result from the observations, gives a number containing more than the four necessary digits. Thus, if we take seven observations of a quantity, each to three figures, and take the mean, we shall usually get any number of digits we please when we divide by the 7. But we know that the observations are only accurate to three figures; hence, in the mean obtained, all the figures after the fourth, at any rate, have no meaning. They are introduced simply by the arithmetical manipulation, and it is, therefore, better to discard them. It is, indeed, not only useless to retain them, but it may be misleading to do so, for it may give the reader of the account of the experiment an impression that the measurements have been carried to a greater degree of accuracy than is really the case. Only those figures, there- fore, which really represent results obtained by the measure- ments should be included in the final number. In dis- carding the superfluous digits we must increase the last digit retained by unity, if the first digit discarded b 5 or greater than 5. Thus, if the result of a division gives 32*316, we adopt as the value 32*32 instead of 32*31. For it is evident that the four digits 32*32 more nearly re- present the result of the division than the four 32*31. Superfluous figures very frequently occiur in the multi- plication and division of approximate values of quantities. These have also to be discarded from the result ; for if we multiply two numbers, each of which is accurate only to one part in a thousand, the result is evidently only accurate to the same degree, and hence all figures after the fourth must be discarded. The arithmetical manipulation may be performed by using logarithms, but it is sometime -^ctically shorter to work out the arithmetic than to bgarithms ; and in this case the arithmetical process ,.^ oe much a iated by discarding uimecessary figures in the co' of the work. 38 Practiced Physics. [Chap. IIL The following examples will show how this is managed: — Example (i).— Multiply 656-3 by 4-321 to four figures. Ordinarv form 656-3 4*321 6563 13126 19689 26252 Abbreviated 656-3 4-321 (6563 X 4) « 2625*2 (656x3) - 1968 (65x2) - 13-0 (6x1) - 6 2835-8723 Result 2836 2835-6 Result 2836 The multiplication in the abbreviated form is conducted in the reverse order of the digits of the multiplier. Each successive digit of the multiplier begins at one figure further to the left of the multiplicand. The decimal point should be fixed when the multiplication by the first digit (the 4} is completed. To make sure of the result being accurate to the requisite number of places, the arithmetical calculation should be carried to one figure beyond the degree of accuracy ultimately r^uired. Exctmpie (2). — Divide 65-63 by 4-391 to four figures. 4*390 6563000 (14946 Abbreviated form 4*390 65-630 (14948 4391 4391 21720 17564 21720 17564 •41560 (439) 4156 39519 3951 -20410 17564 (43) -205 172 •2846 (4) -33 Result 14-95 Result 14-95 In the abbreviated form, instead of performing the successive steps of the division by bringing down o's, sue- Chaf. hi.] Physical Arithmetic, 39 cessive figures are cut off from the divisor, beginning at the right hand ; thus, the divisors are for the first two figures of the quotient 4391 ; for the next figure, 439 ; for the next, 43- It can then be seen by inspection that the next figure is 8. The division is thus accomplished. It will be seen that one o is added to the dividend ; the arithmetic is thus carried, as before, to one figure b^ond the accuracy ultimately required This may be avoided if we always multiply the divisor mentally for one figure beyond that which we actually use, in order to determine what number to 'carry'; the number carried appears in the work as an addition to the first digit in the multipli- cation. The method of abbreviation, which we have here sketched, is especially convenient for the application of small corrections (see below, p. 42). We have then, gene- rally, to multiply a number by a factor differing but little from unity ; let us take, for instance, the following : — Example (3). — Multiply 563*6 by 1*002 to four places of decimals. Adopting the abbreviated method we get— 563*6 1*002 563^ 1*1 5647 Result 5647 or Example (4).— Multiply 563*6 by -9998. In this case -9998 - 1 - -0002. 563-6 I— •0002 563^ 562*5 Result 562*5 46 Practical Physics. [Chap. IIL It will be shewn later (p. 44) that dividing by '9998 is the s?" i, as far as the fourth place of decimals is concerned, as multiplying by 1-002, and vice versd\ this suggests the possibility of considerable abbreviation of arithmetical cal- cu)^on in this and similar casea Facilitation of Arithmetical Calculation by means of Tables, — Interpolation. The arithmetical operations of multiplication, division, the determination of any power of a number, and the ex- traction of roots, may be performed, to the required degree of approximation, by the use of tables of logarithms. The method of using these for the purposes mentioned is so well known that it is not necessary to enter into detsdls here. A table of logarithms to four places of decimals is given in Lupton's book, and is sufficient for most of the calculations that we require. If greater accuracy is necessary, Cham- bers's tables may be used. Instead of tables of logarithms, a * slide-rule ' is sometimes employed. An explanation o\ the plan upon which the rule is graduated and the method of using it for making arithmetical calculations is given at the end of this chapter, pp. 51-58. Besides tables of logarithms, tables of squares, cubes, square roots, cube roots, and reciprocals may be used Short tables will be found in Lupton's book (pp. 1-4); foi more accurate work Barlow's tables should be used. Besides these the student will require tables of the trigono- metrical functions, which will also be found among Lupton's tables. An arithmetical calculation can frequently be simplified on account of some special peculiarity. Thus, dividing by 5 is equivalent to multiplying by 2, and moving the decimal point one place to the left. Again, ir* = 9'87 = 10— '13, and many other instances might be given ; but the student can only make use of such advantages by a familiar acquaint- ance with cases in which they prove of service. Chap. III.] Physical Arithmetic. ^%\ In some cases the variations of physical quantities are also tabulated, and the necessity of performing t"'? arith- metic is thereby saved. Thus, No. 31 of Lupton^is tables gives the logariUims of (i + -00367 i) for successive degrees of temperature, and saves calculation when the volur*>e or pressure of a mass of gas at a given temperature is required A table of the variation of the specific resistance of copper with variation of temperature, is given on p. 47 of tfie same work. It should be noticed that all tables proceed by certain definite intervals of the varying element ; for instance, for successive degrees of temperature, or successive units in the last digit in the case of logarithms ; and it may happen that the observed value of the element lies between the values given in the table. In such cases the required value can generally be obtained by a process known as Mnterpolation.' If the successive intervals, for which the table is formed, are small enough, the tabulated quantity may be assumed to vary uniformly between two successive steps of the varying element, and the increase in the tabulated quantity may be calculated as being proportional to the increase of the vary- ing element We have not space here to go more into detail on this question, and must content ourselves with say- ing that the process is strictly analogous to the use of * pro- portional parts' in logarithms. We may refer to §§ 12, 19, 77 for examples of the application of a somewhat analogous method of physical interpolation. Algebraical Approximation, Approximate Formula. Introduction of small Corrections, If we only require to use a formula to give a result accurate within certain limits, it is, in many cases, possible to save a large amount of arithmetical labour by altering the form of the formula to be employed This is most frequently the case when any smaU correction to the value of one of the observed elements has to be introduced, as in the case, for instance, of an observed barometric height which has to 42 Practical Physics. [Chap. III. be corrected for temperature. We substitute for the strictly accurate formula an approximate one, which renders the calculation easier, but in the end gives the same result to the required degree of accuracy. We have already said that an accuracy of one part in a thousand is, as a rule, ample for our purpose ; and we may, therefore, for the sake of definiteness, consider the simplifi- cation of algebraical formula with the specification of one part in a thousand, or o*i per cent, as the limit of accuracy desired. Whatever we have to say may be easily adapted for a higher degree of accuracy, if such be found to be necessary. It b shewn in works on algebra that (i + ^)" = I + « ;ip + -^^ 'x^ + terms involving higher powers of * (i). This is known as the ' binomial theorem/ and is true for all values of n, positive or negative, integral or fiac- tionaL^ Some special cases will probably be familiar to every student, as : — (l+jir)-> = -I- = i-;t + ^«-;c»+ .... l-f ^ If we change the sign of x we get the general formula in the form ' 2 We may include both in one form, thus : — (i±jc)«= i±«ar+^^^:::i)*»± .... ' 2 where the sign + means that either the + or the — is to be taken throughout the formula. * If n be negative or a fraction then x must be less than unity. Chap. iiL] Physical Arithmetic 43 NoWj if ^ be a smaU fraction, say, i/iooo or cooi, o^ is evidently a much smaller fraction, namely, 1/1000,000, or o'oooooi, and 0^ is still smaller. Thus, unless n is very large indeed, the term 2 *» will be too small to be taken account of, and the terms which follow will be of still less importance. We shall probably not meet with formula in which n is greater than 3. Let us then determine the value of x so that 2 may be equal to *ooi, that is to say, may just make itself felt in the calculations that we are now discussing. Putting « = 3 we get 3 .T* = 'OOI X = -/ 00033 = '02 roughly. So that we shall be well within the truth if we say that (when If = 3), if ^ be not greater than 0*01, the third term of equation (i) is less than "ooi, and the fourth term less than 'ooooi. Ndther of these, nor anyone beyond them, will, therefore, affect the result, as far as an accuracy of one part in a thousand is concerned ; and we may, therefore, say that, if ^ is not greater than 0*01, (l+*)»=:l+3X To use this approximate formula when ^ = 0*01 would be inadmissible, as it produces a considerable effect upon the next decimal place ; and, if in the same formula, we make other approximations of a similar nature, the accumulation of approximations may impair the accuracy of the result In any special case, therefore, it is well to consider 44 Practical Physics. [Chap. IIL whether x is small enough to allow of the use of the approxi- mate formula by roughly calculating the value of the third term ; it is nearly always so if it is less than '005. This in- cludes the important case in which x is the coefficient of expansion of a gas for which x = '00367. If /I be smaller than 3, what we have said is true within still closer limits ; and as » is usually smaller than 3, we may say generally that, for our purposes, and (i— ^)"= I— «x, provided x be less than 0*005. Some special cases of the application of this method of approximation are here given, as they are of frequent occur- rence : — (i±^)'= I ±2* (i±jc)'= i±3jc ^i±x^{i±x)\ = i±- I ^===(i±^)-J=izp- = (l±JC)-*=r 1:^2*. The formulae for +^ and ^x are here included in one expression ; the upper or lower sign must be taken through- out the formula. We thus see that whenever a factor of the form (i ±.xf occurs in a formula where « is a small fraction, we may replace it by the simpler but approximate factor i+n jr; and we have already shown how the multiplication by such a factor may be very simply performed (p. 39). Cases of the application of this method occur in §§ 13, 24 etc Another instance of the change of formula for the pur- Chap. III.] Physical Arithmetic 45 poses of arithmetical simplicity is made use of in § 13. In that case we obtain a result as the geometric mean of two nearly equal quantities. It is an easy matter to prove algebraiodly, although we have not space to give the proof here, that the geometric mean of two quantities which differ only by one part in a thousand differs from the arithmetic mean of the two quantities by less than the millionth of either. It b a much easier arithmetical operation to find the arithmetic mean than the geometric, so that we substi- tute in the formula (x'\-x')/2 for \/;c x'. The calculation of the effect upon the trigonometrical ratios of an angle, due to a small fractional increase in the angle, may be included in this chapter. We know that sin (O+d) = sin tf cos ^-h cos 6 sin d. Now, reference to a table of sines and cosines will shew that cos d differs from unity by less than one part in a thousand if ^be less than 2*^ 33', and, if expressed in circular measure^ the same value of d differs from sin^ by one part in three thousand; so we may say that, provided d is less than 2 J**, cos d is equal to unity, and sin^ is equal to d expressed in circular measure. The formula is, therefore, for our purposes, equivalent to sin (tf +^ = sin tf +^cos A We may reason about the other trigonometrical ratios in a similar manner, and we thus get the following approximate formulae : — sin {B±d) =s sin ^±//cos A cos (p±-d) = cos dqi^sin ft tan \0±.d) = tan 6±d sec « A The upper or lower sign is to be taken throughout the formula. If // be expressed in degrees, then, since the circular 46 Practical Physics: [Chap, in, measure of i^ is ir/i8o^ that of if' is iir/iSo, and the formulae become sin {e±d) = sin tf ±^cos 0, loO &C. It has been ahready stated that approximate formulae are frequently available when it is required to introduce correc- tions for variations of temperature, and other elements which may be taken from tables of constants. There is besides another use for them which should not be overlooked, namely, to calculate the effect upon the result, of an error of given magnitude in one of the observed elements. This is practically the same as calculating the effect of a hypothe- tical correction to one of the observed elements. In cases where the formula of reduction is simply the product or quotient of a number of factors each of which is observed directly, a fractional error of any magnitude in one of the factors produces in the result an error of the same frac- tional magnitude, but in other cases the effect is not so simply calculated If we take one example it will serve to illustrate our meaning, and the general method of employ- ing the approximate formulae we have given in this chapter. In § 75 electric currents are measured by the tangent galvanometer. Suppose that in reading the galvanometer we cannot be sure of the position of the needle to a greater accuracy than a quarter of a degree. Let us, there- fore, consider the following question : — * What is the effect upon the value of a current^ as deduced from observations with the tangent galvanometer^ of an error of a quarter of a degree in the reading ? ' The formula of reduction is Suppose an error 8 has been made in the reading of tf, »o that the observed value is = i6(tand+8sec»^. . . . (p. 45) Chap. III.] Physical Arithmetic 47 The fractional oror g in the result is c ktSLTiO smTcosU 23 sin 2 6^ The error S must be expressed in circular measure ; if it be equivalent to a quarter of a degree, we have • • The actual magnitude of this fraction depends upon the value of 0, that is upon the deflection. It is evidently very great when is very small, and least when a= 45^, when it is o'9 per cent From which we see not only that when is known the effect of the error can be calculated, but also that the effect of an error of reading, of given magnitude, is least when the deflection is 45*". It is clear from this that a tangent galvanometer reading is most accurate when the deflection produced by the current is 45*". This furnishes an instance, therefore, of the manner in which the approxi- mate formulae we have given in this chapter can be used to determine what is the best experimental arrangement of the magnitudes of the quantities employed, for securing the greatest accuracy in an experiment with given apparatus. The same plan may be adopted to calculate the best arrangement of the apparatus for any of the experiments described below. In concluding this part of the subject, we wish to draw special attention to one or two cases, already hinted at, in which either the method of making the experiments, or the formula for reduction, makes it necessary to pay special attention to the accuracy of some of the elements observed. In illustration of the former case we may mention the weighing of a small mass contained in a large vessel To 48 Practical Physics. [Chap. hi. fix ideas on the subject, consider the determination of the mass of a given volume of gas contained in a glass globe, by weighing the globe full and empty. During the interval between the two weighings the temperature and pressure of the air, and in consequence the apparent weight of the glass vessel, may have altered. This change, unless allowed for, will appear, when the subtraction has been performed, as an error of the same actual magnitude in the mass of the gas, and may be a very large fraction of the observed mass of the gas, so that we must here take account of the variation in the correction for weighing in air, although such a precaution might be quite unnecessary if we simply wished to determine the actual mass of the glass vessel and its contents to the degree of accuracy that we have hitherto assumed. A case of the same kind occurs in the determination of the quantity of moisture in the air by means of drying tubes (§ 42). Cases of the second kind referred to above often arise from the fact that the formulae contain diffeiences of nearly equal quantities ; we may refer to the formulas employed in the correction of the first observations with Atwood's machine (§ 21), the determination of the latent heat of steam (§ 39), and the determination of the focal length of a concave lens (§ 54) as instances. In illustration of this point we may give the following question, in which the hypothetical errors introduced are not really very exaggerated. * An observer, in making experiments to determine the focal length of a concave lens, measures the focal length of the auxiliary lens as 10*5 cm., when it is really 10 cm., and the focal length of the combination as 14*5 cm., when it is really 15 cm. ; find the error in the result introduced by the inaccuracies in the measurements' We have the formula Chaf. III. J Physical Arithmetic 49 whence putting in the true values of f and/|. ' IS -10 5 ^^ and putting the observed values /, = -i£5Xi?:5 ^«£5£25 =-38.06. 145 -105 4 The fractional error thus introduced is 806 or more than- 35 per cent, whereas the error in either observation was not greater than 5 per cent It will be seen that the large increase in the percentage error is due to the fact that the difference in the errors in F and/i has to be estimated as a fraction of f— /i ; this should lead us to select such a value of /i as will make F— yi as great as possible, in order that errors of given actual magnitude in the observations may produce in the result a fractional error as small as possible. We have not space for more detail on this subject The student will, we hope, be able to understand from the in- stances given that a large amount of valuable information as to the suitability of particular methods, and the selection of proper apparatus for making certain measurements, can be obtained from a consideration of the formulae of reduc- tion in the manner we have here briefly indicated. Graphical Methods, The results of a large number of experiments can be best expressed graphically. Examples of this method will be E 50 Practical Physics, [Chap III. found in the course of the book. (See specially §§ 26, 40, 41.) The method is chiefly useful in cases in which we wish to trace the dependence of one quantity on another. Paper suitable for the purpose, ruled in small squares, can be easily obtained. In applying the method, the values of the independent variable are set down as abscissae parallel to one set of lines, the corresponding values of the dependent variable being measured as ordinates at right angles to this. In cases in which the phenomenon under investigation is continuous in its character, a smooth curve can usually be drawn, either freehand or by the aid of a flexible ruler, so as to pass approximately through these points, and the law sought can be obtained by an investigation of the form of the curve. Thus, suppose we are endeavouring to prove that the pressure of a given mass of gas at constant volume varies as the absolute temperature, we lay ofl" as abscissae the observed values of the temperature, say in degrees centigrade from freezing point as zero, and as ordinates the correspond- ing pressures. On drawing the curve which best represents the experi- ments we And it to be a straight line ; moreover, this line cuts the line of np pressure from which the ordinates are measured at a point on the negative side of the origin about 273** C. below freezing point This point is the absolute zero, and the pressure is clearly proportional to the temperature reckoned from it. The accuracy of a result obtained by a graphical method will, to some extent, depend on the scale adopted. Let us suppose that in the above experiment we can read the temperature to o-i^'C, and the pressure to -5 mm. Then it is clear we must adopt such a scale for the tempe- rature, if we wish to be accurate, as will allow o'l^C Chap. Ill.l Physical Arithmetic, 51 to be clearly visible. We might take i inch to repre- sent I**. If at the same time we represent i cm. of pressure by I inch on the diagram, we can plot down the pressure to *5 mm., and these scales will give us satisfactory results. The figure so drawn will be very large, larger than is required for the accuracy attempted in most of the experi- ments described. When the diagram is to be used to represent the varia- tions of one quantity corresponding to those of another over a small range, a wide scale can be used without making a very large diagram by using the abscissse or ordinates, 01 both, to represent the respective changes and not the whole quantities. Thus, suppose we wish to represent the changes of volume of one gramme of water consequent on changes of temperature between o** C. and 10® C. ; we may regard the horizontal line through the origin as indicating volumes equal to that of one gramme of water at 4^ C, and one inch of vertical height may represent a change of volume of '0000 1 C.C. The line of no volume would, if drawn, be 100,000 inches below the horizontal through the origin. But it need not be drawn ; and if one inch of horizontal distance represent i** C, the whole diagram will be com- prised in a space 10 inches square. In drawing a diagram the horizontal and vertical scales chosen should always be very clearly set out in the diagram itself. The Slide RuU. The slide rule is a mechanical contrivance for perform- ing rapidly various arithmetical operations. Its action depends in the main on the two principles that the loga- rithm of the product of two numbers is the sum of the logarithms of its factors, and that the logarithm of the «th power of a number is n times the logarithm of the number. B a 52 Practical Physics, (Cuaf. m In its very simplest form a slide rule would consist of two identical scales, one of which can slide along the other. The scales are divided in such a way that the distance along either scale measured from one end — say, the left- hand — is proportional to the logarithm of the corresponding scale number. Thus the distance from the left-hand end to a reading a, say, is proportional to the logarithm of a ; that to a second reading h is proportional to the logarithm of^. One of the two scales is known as the rule ; the other as the slider. Now let p be the mark on the rule corresponding to a division a, a being the index at the left-hand end of the rule, then A p measures the logarithm of a, so that the number at A is I. Place c, the index of the slider, which is marked i, in contact with p, and let Q be the mark on the slider which corresponds to a division ^, so that CQ measures log b. Let R be the mark on the rule opposite Q, let r be the corresponding reading ; then a r = log c. Now log a ^ = log a + log ^ = AP + CQ = AP + PR K ar as logr; .*. c = ab. In the figure as drawn, if the distance a b be taken as f J r f [ f T[T 1 — I [ I i 1 1 5 4 9678010 k ^ unity, then a P is log 3, Q is at division 3 on the slider, and R, the corresponding division on the scale, is 9, which is equal to 3 times 3. The above result, then, leads to the following method for obtaining the product of two or more quantities by the slide rule : — ^Thus, if a and h are the quantities, set the index of Chap, in.] Physical A rtfhnutie. 5 3 the slider to division a oa the rule, and read the division of the rule which corresponds to division b of the slider. This gives the product a b. The inverse of this gives us the method of division. Thus, to divide c by b, set divi- sion b of the slider opposite c of the rule, and read the division a, say, of the rule opposite the index of the slider ; then, clearly, a is equal to cjb. If, as b usually the case, the scale and the slider are of the same length, it will often happen that when the index of the slider is set to a division a^ the division of the rule which corresponds to b on the slide is off the scale. The following considerations will shew us how to proceed in this case. Let us suppose the rule is divided into ten parts, marked I, 2, to 10, each of these being subdivided into tenths or twentieths. These subdivisions may be still further divided by eye to fifths, so that we read with fair accuracy to *oi. The divisions gradually get smaller as we go up the scale ; in many rules the lower numbers are subdivided to hundredths. Thus the distance measured from the index, or division i, of the scale to a division such as 783 gives us the logarithm of 7*83. Now log 783 = 2 + log 7*83. Thus, to find the logarithm of 783 we have to add 2, that is, twice the length of the scale, to the distance actually given on the scale. We must suppose the scale to be pro- duced backward to the left to twice its own length, and read from this index. Suppose, now, we want to multiply this by 85. The actual distance on the slider up to division 85 is log 8-5. To get log 85 we must add log 10 to this, and log 10 measures the length of the slider. Thus the mark on the slider which we should, according to the rule, put into coincidence with 783 would be at a distance equal to the length of the slider to the left of the index. The complete rule then would consist of a series of repeti- tions of the scales of both rule and slider, the first scale giving logarithms of numbers from i to 10, the next of 54 Practical Physics. [Chap. III. numbers from lo to loo, and so on, and all the scales being exactly alike. Now let us suppose the index of the slider (marked i) to be in coincidence with a division, say 7*80, of the scale ; then 10 on the slider will coincide with 780, 100 of the slider with 780, and so on. Also, since 7*8 x 8*5 is equal to 66'3, we shall find that 8*5, 85, 850, &c., of the slider coincide with 66*3, 663, and 6630 respectively. Thus in multiplying two numbers together it is imma- terial, except so far as the decimal points are concerned, which series of divisions on the rule or slider we use. We may set either division i or division 10 or division 100 of the slider to coincide with one of the given numbers, and look for the number on the rule which coincides with the second number read on the slider. This, with the decimal point inserted in the proper place, will be the product required. If when the index (division i) of the slider is made to coincide with a given division a of the rule, the division b of the slider is off the rule, we must put 10 of the slider to coincide with a, and read the coincidence with ^, which will then be on the scale. This number, with the decimal point properly placed, will be the product a b. To use a slide rule to obtain a square or square root we require two logarithmic scales, one of these being double the length of the other, and the shorter scale being re- peated. In the Gravet form of rule made in celluloid, as supplied by Messrs. Davis & Son, of Derby, the two scales are placed parallel to each other, and the slider moves between them. The slider also carries two scales, the counterparts of those on the rule. The lower scale, which is 25 cm. long, gives a scale of logarithms from i to 10. The left-hand half of the upper scale, 12-5 cm. long, gives a scale of logarithms from i to 10 of half the dimensions adopted for the lower scale. The right-hand half is an exact copy of this, and gives, there- Chap. J II.] Physical A rithmetic. 5 5 fore, when measuring from the index of the first scale, the logarithms of numbers from 10 to 100. A certain length measured on the lower scale gives the logarithm of a number a, say. The same length measured along the upper scale is 2 log a, for the unit of measure- ment of the upper scale is half that of the lower, also 2 log a = log a\ Thus, to find the square of a number, look out the number on the lower scale, and take the reading on the upper scale which coincides with that found on the lower. In order to determine the coincidence, a metal slide, called the Cursor, is employed. This is equivalent to a straight-edge at right angles to the length of the scale which can slide along the scale, and thus facilitates the reading of the coincidences. The rule can be used to find the area of a circle of given radius in the following way : — ^The area of a circle of radius r is x r^. The value of log w (log 3142) is marked on the slide. Set this to the index of the upper scale. Set the cursor to the value of r on the lower scale, and note the reading on the upper scale. This corresponds to log f^. Take the reading qp, the upper scale of the slide which coincides with this, and we obtain the value of V /-*. The cursor may be also used to obtain a continuous product without noting the intermediate steps in the following way : — ^To multiply «, ^, c together, read a on the rule ; set the zero of the slide to this ; set the cursor to ^ on the slide. Move the slide until its index coin- cides with the cursor, and read c on the slide. The corresponding division on the rule gives the value of the product The reverse side of the slider in the rule described contains three scales. One of these is a scale of sines, the 56 Practical Physics. [Chap. III. second a scale of tangents. These are so divided that when either of them is brought into coincidence with the corresponding scale on the rule, the divisions of the rule give respectively the sine or tangent of the angle read on the slider scale. The upper scale of the rule is used for sines, the lower for tangents. The third scale is one of equal parts, and from it the logarithm of a number can be de- tennined. For set this scale so that its zero coincides with the index of the lower scale of the rule, and read any number, 17, say, on this scale. Then, since the distances of the divisions from the index of the scale are proportional to the logarithms of the corresponding numbers, and the whole length of the scale contains lo divisions, we have the ratio log a : log lo = distance of a from end : whole length of scale. Set the cursor to division a, and take the corresponding reading on the scale of equal parts ; let it be x divisions. Suppose that the whole length contains d divisions ; then, since log lo = i, log a = x\d. In the rule already referred to // = 500, so that loga = 2 jc/iooo. This rule also contains a device whereby the logarithms sines, and tangents may be read without reversing the slider. On the under side of the right-hand end of the scale there is a small opening, on each side of which an index mark is seen. When the index of the scale of equal parts, or of sines or tangents coincides with these index marks, it will be found that the scales on the upper side of the rule and slider are coincident. Now draw out the slider, and note the reading a on the Chap. III.J Physical Arithmetic. 57 lower scale with which its index coincides. Note also the reading x on the scale of equal parts. This last reading gives us the distance the slider has moved — ^that is, the distance between the index of the lower scale and the mark a ; but this distance is proportional to log Oj and we have, as before, log a/log 10 = x/d, dT being the number of divisions on the scale of equal parts which correspond with the full length of the logarithmic scale. An exactly similar method applies to finding sines or tangents. The accuracy obtainable with a slide rule depends partly on the exactness with which it is divided, partly on the possible accuracy of setting. Under favourable circumstances an accuracy of i part in 500 is claimed for the rule we have been describing, but this varies in different parts of the scale. Thus suppose we wish to use the rule to multiply 9'22 by, say, 8*53. There are no divisions between 9*2 and 9*25, and the actual distance between these divisions is about 75 mm. To set the slider to this so that the error in the result may be i part in 500, we have to estimate to about one-iiflh of the distance between the marks, or say '15 mm. To do this requires considerable care and practice. Then, again, we have no mark on the slider between 8*5 and 8*55. We have to judge by eye the position of 8*53, and also the division on the scale which coincides with this. The cursor is of help in this, and it is easy to see that the division required lies between 78*5 and 79. Dividing the distance between these divisions by eye with the aid of a magnifying glass, we get as the result 78*6 . . . , and the last figure will be certainly right to i, which is about i in 800 in the result As another example, suppose we wish to find the circumference of a circle 1752 inches in diameter. To 58 Practical Physics. [Chap. IIL read the last figure correctly on the scale we have to sub- divide to tenths a distance of about *5 mm. ; but an error of 2 in this figure, with a corresponding error in the value of IT (3-1416), will only affect the result to i part in 500. There are no divisions between 3*14 and 3*15, but the distance between these two can be subdivided into fifths, and we can set the cursor to 3'i42, correct to •002. The product lies between 5*50 and 5*55, and this dis- tance, which is well over i mm. in length, can be subdivided to fifths with certainty. We obtain as the result 5*51, the true value being 5 '504. Or, again, find the angle whose sine is '8. The divisions in the neighbourhood of 8 on the upper 5cale, which is used here, are about 75 mm., and we can set the scale with fair accuracy. The angle is seen to be between 53° and 54*". To get it more nearly we have to divide a distance of about a millimetre into parts. We can do this to fifths or sixths, giving an accuracy of^ say, 10 minutes, or i in 300. For angles above 60** the degree divisions on the scale of sines are very small, while between 70® and 80*^ each division is 2**, and the divisions corre- sponding to 80^ and 90^ are only about i mm. apart. The \^ue of sin a changes by about i per cent for 1° when a is about 60**, and the setting can be done to about one-fifth or one-sixth of a degree in this position. Thus it will be seen that with care the accuracy of nearly i in 500 is attainable over a wide range. Ch. IV. § I.] Measurement of the Simple Quantities, 59 CHAPTER IV. MEASUREMENT OF THE MORE SIMPLE QUANTITIES. LENGTH MEASUREMENTS. The general principle which is made use of in measuring lengths is that of direct comparison (see p. 2); in other words, of laying a standard, divided into fractional parts, against the length to be measured, and reading ofif from the standard the number of such fractional parts as lie between the extremities of the length in question. Some of the more important methods of referring lengths to a standard, and of increasing the accuracy of readings, may be exemplified by an explanation of the mode of using the following instruments. I. The Calipers. This instrument consists of a straight rectangular bar of brass, d b (fig. i), on which is engraved a finely-divided scale. From this bar two steel jaws project These jaws are at right angles to the bar ; the one, d f, is fixed, the other, c o, can slide along the bar, moving accurately parallel to itself. The faces of these jaws, which are opposite to each other, are planed flat and parallel, and can be brought into contact On the sliding piece c will be observed two shoit scales called verniers, and when the two jaws are in contact, one end of each vernier, marked by an arrowhead in the figure, coincides with the end of the scale on the bar.* If then, in any other case, we determine the position of this end of the vernier with reference to the scale, we find the distance between these two flat faces, and hence the length of any object which fits exactly between the jaws. It will be observed that the two verniers are marked ' out- sides and ' insides ' respectively.^ The distance between the ' If with the instrument employed this is found not to be the case, t correction must be made to the observed length, as described in J ^ k similar remark applies to § 2. ' See frontispiece, fig. 3. to Practical Physics, ECh. IV. § 1. Fig. I. jaws will be ^ven by the outsides vernier. The otner pair of faces of these two jaws, opposite to the two plane parallel ones, are not plane, but cylindrical, the axes of the cylinders being also perpendicular to the length of the brass bar, so that the cross section through any point of the two jaws, when pushed up close together, will be of the shape of two U*s placed opposite to each other, the total width of the two being exactly one inch. When they are in contact, it will be found that the arrowhead of the vernier attached to the scale marked insides reads exactly one inch, and if the jaws of the calipers be fitted inside an object to be mea- sured — e.g., the internal dimensions of a box — the reading of the vernier marked insides gives the distance required. Suppose it is required to measure the length of a cylinder with fiat ends. The cylinder is placed with its axis parallel to the length of the calipers. The screw a (fig. i) is then turned so that the piece A attached to it can slide freely along the scale, and the jaws of the calipers are adjusted so as nearly to fit the cy- linder (which is shown by dotted lines in the diagram). The screw a is then made to bite, so that the attached piece is ' clamped ' to the scale Another screw, b, on the under side of the scale, will, if now turned, cause a slow motion of the jaw c o, and by means of this the fit is made as accurate as possible. This is considered to be attained when the cylinder is just held firm. This screw b is called the * tangent screw,* and the adjustment is known as the ' fine adjustment' It now remains to read upon the scale the length of the cylinder. On the piece c will be seen two short scales — the * outsides ' and * insides ' already spoken of These short scales are called 'verniers.' Their use is to increase the Mi III II II ]4 Ch. IV. § i.J Measurement of the Simple Quantities. 61 accuracy of the reading, and may be explained as follows : suppose that they did not exist, but that the only mark on the piece c was the arrowhead, this arrowhead would in all probability lie between two divisions on the large scale. The length of the cylinder would then be less than that corresponding to one division, but greater than that corre- sponding to the other. For example, let the scale be actually divided into inches, these again into tenths of an inch, and the tenths into*five parts each; the small divisions will then be ^ inch or '02 inch in length. Suppose that the arrowhead lies between 3 and 4 inches, between the third and fourth tenth beyond the 3, and between the first and second of the five small divisions, then the length of the cylinder is greater than 3H-nF+A> '-^ >3*3* inches, but less than 3+A+yV *-^ <3'34 inches. The vernier enables us to judge very accurately what fraction of one small division the distance between the arrowhead and the next lower division on the scale is. Observe that there are twenty divisions on the vernier,^ and that on careful ex- amination one of these divisions coincides more nearly than any other with a division on the large scale. Count which division of the vernier this is — say the thirteenth. Then, as we shall show, the distance between the arrowhead and the next lower division is \% of a small division, that is ,j.^^s*oi3 inch, and the length of the cylinder is therefore 3+A+ir(r+Ti^=3'32 + -oi3=:3-333 inch. We have now only to see why the number representing the division of the vernier coincident with the division of the scale gives in thousandths of an inch the distance between the arrowhead and the next lower division. Turn the screw-head b till the arrowhead is as nearly coincident with a division on the large scale as you can make it. Now observe that the twentieth division on the vernier is coincident with another division on the large scale, and that the distance between this division and the first is nineteen small divisions. Observe also that no other ' Various forms of vernier are figured in the frontispiece. 62 Practiced Physics. [Ch. IV. § i. divisions on the two scales are coincident Both are evenly divided ; hence it follows that twenty divisions of the vernier are equal to nineteen of the scale— that is, one division on the vernier is ^ths of a scale division, or that one division on the vernier is less than one on the scale by j^th of a scale division, and this is n^i^th of an inch.' Now in measuring the cylinder we found that the thirteenth division of the vernier coincided with a scale divi- sion. Suppose the unknown distance betwefen the arrowhead and next lower division is x. The arrowhead is marked o on the vernier. The division marked i will be nearer the next lower scale-division by Tinny^^ ^^ ^"^ vm^^ for a vernier division is less than a scale division by this amount. Hence the distance in inches bet^veen these two divisions, the one on the vernier and the other on the scale, will be The distance between the thirteenth division of the vernier and the next lower scale division will similarly be But these divisions are coincident, and the distance between them is therefore zero ; that is ^=t^* Hence the rule which we have already used. The measurement of the cylinder should be repeated four times, and the arithmetic mean taken as the final value. The closeness of agreement of the results is of course a test of the accuracy of the measurements. The calipers may also be used to find the diameter of the cylinder. Although we cannot here measure surfaces which are strictly speaking fiat and parallel, still the portions of the surface which are touched by the jaws of the calipers are very nearly so, being small and at opposite ends of a diameter. Put the calipers on two low supports, such as a pair of glass rods of the same diameter, and place the cylinder on end upon the table. Then slide it between the jaws of the » GeneraUy, if n divisions of the vernier arc equal to n^\ of the scale, then the vernier reads to i/wth of a division of the scale. Ch. rv. § 2,] Measurement of tJu Simple Quantities, 63 calipers, adjusting the instrument as before by means of the tangent screw, until the cylinder is just clamped Repeat this twice, reading the vernier on each occasion, and taking care each time to make the measurement across the same diameter of the cylinder. Next take a similar set of readings across a diameter at right angles to the former. Take the arithmetic mean of the different readings, as the result Haying now found the diameter, you can calculate the area of the cross section of the cylinder. For this area is -— , d being the diameter. The volume of the cylinder 4 can also be found by multiplying the area just calculated by the length of the cylinder.* Experiments. Determine the dimensions (i) of the given cylinder, (2) ol the given sphere. Enter results thus : — X. Readings of length of cylinder, of diameter. 3-333 in. j^.^^ ^ 1 1-301 in. 3332 >, lr303 » 3*334 » Djam. 2 3'334 „ Mean 3*3332 « Mean 1*3022,, Area - 1*3318 sq. in. Volume - 4*4392 cu. in, 2. Readings of diameter of sphere. Diam. 1 5*234 in. » 2 5*233 „ >» 3 5*232 >» ». 4 5233 H Mean 5*233 „ 3. The Beam-Compass. The beam-compass, like the calipers, is an instrument for measuring lengths, and is very similar to them in con- struction, consisting essentially of a long graduated beam * A cylinder whose volume has been thus determined can be used to find Ihe true density of water in grammes per cc The additional obser- vations required art the weight of the cylinder in vacuo and in water. \ *f ^ If 1 1 -303 » 1 1-302 „ 64 Practical Physics. [Ch. IV. § a. with one steel compass-point fixed at one end of it, and another attached to a sliding piece provided with a fiducial mark and vernier. These compass-points take the place of the jaws of the calipers. It differs from them however in this, that while the calipers are adapted for end-measures such as the distance between the two flat ends of a cylinder, the beam-compass is intended to find the distance between two marks on a flat sur&ce. For example, in certain experiments a paper scale pasted on a board has been taken to represent truly the centimetres, millimetres, &c marked upon it We now want to know what error, if any, there is in the divisions. For this purpose the beam-compass is placed widi its scale parallel to the paper scale, and with the two compass points lying in a convenient manner upon the divisions. It will be found that the beam-compass must be raised by blocks of wood a little above the level of the paper scale, and slightly tilted over till the points rest either just in contact with, or just above, the paper divisions. One of the two points is fixed to the beam of die com- pass ; we will call this a. The other, b, is attached to a sliding piece, which can be clamped by a small screw on a second sliding piece. First unclamp this screw, and slide the point B along, till the distance a b is roughly equal to the dis- tance to be measured. Then clamp b, and place the point k Fio. t. (^& *) exactly on one of the marks. This is best effected by gende taps at the end of the beam with a small mallet St4tiiy "* [ \ ' It is the inside edge of the compass- point which has to be brought into co- incidence with the mark. Now observe that, although b is clamped it is capable of a slow modon by means of a second screw called a * tangent screw,' whose axis is parallel to the beam. Move this screw, with so light a touch as not to dbturb the position of the beam-compass, until the point B is on the other mark, i.e. the inside edge of b coincides with Ch. IV. § 2.] Measurement of the Simple Quantities, 65 the division in questioa Suppose that the point a is on the right-hand edge of the paper scale division, then b should also be on the right-hand edge of the corresponding division. To ensure accuracy in the coincidence of the edges you must use a magnifying-glass. You have now only to read the distance on the beaui- scale. To do this observe what are the divisions between which the arrowhead of the vernier* Ms. Then the reading required is the reading of the lower of these divisions + the reading of the vernier. The divisions are each i milli- metre. Hence, if the arrowhead falls between the 125th and 126th, the reading is 125 muL + the reading of the vernier. Observe which division of the vernier is in the same straight line with a division of the scale. Suppose the 7th to be so situated Then the reading of the vernier is -^ mm. and the distance between the points is 125*7 mm. Repeat the observation twice, and suppose that 125-6 and 125*7 are the readings obtained, the mean of the three will be 125*66, which may be taken as the true distance between the marks in questioa Suppose that on the paper scale this is indicated by 126 mm., then to make the scale true we must reduce the reading by -34 mm. This is the scale correction for this division. . Experiment, — Check by means of the beam-compass the accuracy of the divisions of the given centimetre scale. Enter results thus : — Division of scale at Division of scale at Vernier readings which A is placed which B is placed (mean of 3 obs.) I cm. 1*005 cm. • 2 „ 2*OIO „ n 3 «i 3010 „ n 4 ,1 4015 .» • • 5 >» etc. 5*015 ^ t See froDtispiece, fig. I. F 66 Practical Physics. [Oil. IV. § 3 3. The Sorew-Oange. This instrument (fig. 3) consists of a piece of solid metal with two arms extending perpendicularly from its two Fig. 3. ends. To the one arm a steel plug, p, with a care- X n P c-^^w^ fully planed face, is fixed, I and through the other arm, opposite to the plug, a screw c passes, having a plane face parallel and opposite to that of the plug. The pitch of the screw is half a millimetre, and consequently if we can count the number of turns and fractions of a turn of the screw from its position when the two plane faces (viz. that of the plug and that of the screw) are in contact, we can determme the distance in millimetres between these two parallel surfaces when the screw is in any position. In order to do this the more conveniently, there is at- tached to the end of the screw farther from the plug a cap x, which slides over the cylindrical bar through which the screw passes ; this cap has a bevelled edge, the circumference of which is divided into fifty equal parts. The circle on the cylindrical bar, which is immediately under the bevelled edge, when the two opposing plane surfaces are in contact, is marked i^ and a line drawn parallel to the length of the cylinder is coincident (if the apparatus is in perfect adjust- ment) with one of the graduations on the bevelled edge which we will call the zero mark of that edge. Along this line a scale is graduated to half-millimetres, and hence one division of the scale corresponds to one complete turn of the cap and screw. Hence the distance between the parallel planes can be measured to half a millimetre by reading on this scale. We require still to determine the fraction of a turn. We know that a complete revolution corresponds to half a millimetre ; the rotating edge is divided into fiftj parts, and Ch. rv. § 3.] Measurement of the Simple Quantities. 67 therefore a rotation through a single part corresponds to a separation of the parallel planes by y^ mm. Suppose, then, that the scale or line along which the graduations on the cylinder are marked, cuts the graduations on the edge of the cap at 1 2 '2 divisions from the zero mark ; then since, when a revolution is complete, the zero mark is coincident with the line along which the graduations are carried on the cylinder, the distance between the parallel planes exceeds the number of complete revolutions read on that scale by -^V ^'^^ ^^ * turn, Le. by '122 mm. If then we number every tenth division on the bevelled edge successively i, 2, 3, 4, 5, these numbers will indicate tenths of a millimetre; 5 of them will be a complete turn, and we must go into the next turn for 6, 7, 8, 9 tenths of a millimetre. It will be noticed that on the scale gradu- ated on the fixed cylinder the smaller scratches correspond to the odd half-millimetres and the longer ones to the com- plete millimetres. And on the revolving edge there are two series of numbers, i, 2, 3, 4, 5 inside, and 6, 7, 8, 9, 10 out- side. A little consideration will shew that the number to be taken is the inside or the outside one according as the last visible division on the fixed scale is a complete millimetre division or an odd half-millimetre division. We can therefore read by this instrument the distance between the parallel planes to y^th of a millimetre, or by estimating the tenth of a division on the rotating edge to the TiAnr^ ^^ ^ millimetre. We may use the instrument to measure the length of a short cylinder thus. Turn the screw-cap, holding it quite lightly, so that, as soon as the two parallel planes touch, the fingers shall slip on the milled head, and accordingly shall not strain the screw by screwing too hard.^ Take a reading when the two planes are in contact; this gives the zero read- * Special jprovirion is made for this in an improved fonn of this apparatus. The milled head is arranged so that it slips past a ratchet wheel whenever the pressure on the screw-face exceeds a certain limit F 2 70 Practical Physics. [Ch. IV. § 4. and each of these subdivided into ten. Let us suppose that division 12 of the disc is opposite to the scale at f, and that the milled head is turned until division 36 comes oppo- site. Then the head has been turned through 24 (i.e. 36 — 1 2) larger divisions ; but one whole turn or fifty divisions carry the point d through \ mm. Thus a rotation through twenty-four divisions will carry it through |J of \ mm. or *24 mm. Hence the larger divisions on the disc r c correspond to tenths of a millimetre, and these are subdivided to hundredths by the small divisions. Thus we might have had opposite to the scale in the first instance 12-6 large divisions, and in the second 36*91 Then the point d would have moved through '243 mm. It will be noticed that in the figure division o is in the centre of the scale h k, which is numbered i, 2, 3, &c, fi-om that point in both directions up and down. The divisions numbered on the disc f g are the even ones * — 2, 4, 6, &c — and there are two numbers to each division. One of these numbers will give the parts of a turn of the screw when it is turned so as to lower the point d, the other when it is turned so as to raise d. Thus in the figure 12 and 38 are both opposite the scale, and in the second position, 36 and 14. We have supposed the head to be turned in such a way that the point d has been lowered through *24 mm. If the rotation had been in the opposite direction, d would have been raised through 0*26 mm. Let us for the present suppose that all our readings are above the zero of the scale. To take a reading we note the division of the scale next above which the disc stands, and then the division of the disc which comes opposite to the scale, taking care that we take the series of divisions of the disc which corresponds to a motion of the point d in the upward direction — the * These numbers are not shewn in the figure. Ch. IV. § 4.] Measurement of the Simple Quantities. 71 inner ring of numbers in the figure. Thus the figured reading is 1*380. If the instrument were in perfect order, the reading when it rested on a plane surface would be o*a This is not generally the case, so we must observe the reading on the plane. This observation should be made four times, and the mean taken. Let the result be '460. Now take the instrument off the plane and draw the middle foot back some way. We will suppose we are going to measure the radius of a sphere from the convex side Place the instrument on the sphere and turn the jcrew B until d touehes the sphere. The position of contact will be given as before, by noticing when the instrument b^;ins to turn round d as a centre. Read the scale and screw-head as before ; let the scale reading be : — 2-5 ; and the disc '235. Then the reading b 2735 mm. Take as before four readings. We require the distance through which the point D has been moved. This is clearly the difference between the two results, or 2*735 — '460 > ^ ^^ <^U ^^ distance a we have a = 2*275 ^'^ It may of course happen that the reading of the instru- ment when on the plane is below the zero ; in this case to find the distance a we must add the two readings. We must now find the distance in millimetres between the feet ab or ac. We can do this directly by means of a finely divided scale ; or if greater accuracy is required, lay the instnmient on a fiat sheet of card or paper, and press it so as to mark three dots on the paper, then measure the distance between these dots by the aid of the beam- compass (§ 2). 72 Practical Physics, [Ch. IV. 5 4. Let us call this length /. Then we can shew* that, if r be the radius required, 6a 2 The observation of / should be repeated about four times. If we wish merely to test if a given surface is spherical, we must measure a for different positions of the apparatus on the surface, and compare the results ; if the surface be spherical, the value of a will be the same for all positions. Experiments, (i) Test the sphericity of the given lens .by observing the value of a for four different positions. (2} Determine the radius of the given sphere for two posi- tions, and compare the results with that given by the calipers. Enter results thus : — Readings on plan« Readings on q>her« 0*460 2735 0463 2733 0458 2734 0'459 2739 Mean 0*460 Mean 2*735 a •2*275 1^1^ Obs. for / 43*56 43*52 43*57 43*59« Mean 43-56. r^ 140*146 mm. By calipers r- 5*517 in. » 140*12 mm. « I Since the triangle fonned by the three feet is equiUtenl, the radius of the circumscribing circle is ,~^-^% *^^—r^ But a beins 2 sin 60® A/3 ^^ the portion of the diameter of the sphere, radius r, cut off by the plane of the triangle, we have (Euc iii 35) /I n ^ tf (2r-tf)-— , whence r«--+ . 3 6a 2 If the dbtance between the centre foot and any one of the three out. side feet be measured, the result is the radius of the circumscribine Mtself. * Ch. IV. § 5.] Measuranent of the Simple Quantities. 73 5. Heasnrement of a Base-Line. The object of this experiment, which is a working model of the measurement of a geodetic base-line, is to determine with accuracy the distance between the scratches on two plugs so fsa apart that the methods of accurate measurement described above are inapplicable. The general plan of the method is to lay ivory scales end to end, fixing them by placing heavy weights on them, and to read by means of a travelling reading microscope the distance between the extreme graduations of the two ivory scales, or between the mark on the plug and the extreme graduation of the ivory scale placed near it We have then to determine the real length of the ivory scales, and by add- ing we get the total length between the plugs. The experiment may therefore be divided into three parts. (i). To determine the Distance between the End Gradu- ations of the Ivory Scales placed end to end This is done by means ofthe travelling microscope. Place the scales with their edges along a straight line drawn between the two marks perpendicular to the scratches, and fix them so that the extreme graduations are within ^th inch. Next place the microscope (which is mounted on a slide similar to the slide-rest of a lathe, and moved by a micrometer screw the thread of which we will suppose is ^th of an inch) so that the line along which it travels on its stand is parallel to the base line, and focus it so ^at one of its cross-wires is parallel and coincident with one edge of the image of the end graduation of the one ivory scale. (It is of no conse- quence which edge is chosen, provided it be always the same in each case.) Read the position of the microscope by its scale and micrometer screw, remembering that the fixed scale along which the divided screw-head moves is graduated to 5oths of an inch, and the circimiference of the screw-head into 74 Practical Physics. [Ch. iv. § 5. 200 parts ; each part corresponds, Aerefore, to ^qioo inch. So that if the reading on the scale be 7, and on the screw- head 152, we get for the position — 7 dirisions of the scale=^in. =0*14 in. 152 divisions of the screw-head =0*0152 in. Reading=50*i552 in. Or if the scale reading be 5 and the screw-head read- ing 15, the reading similarly is 0*1015 in. Next turn the micrometer screw-head until the lost division on the other ivory scale comes into the field of view, and the corresponding edge of its image is coincident with the cross- wire as before. Read again ; the difference of the two readings gives the required distance between the two graduations. In the same way the distance between the scratch on the plug and the end division of the scale may be determined. Place one ivory scale so that one extremity is near to or coincident with the scratch on the plug ; read the dis- tance between them ; then place the other scale along the line and end-on with the first, and measure the distance between the end divisions of the two scales. Then transfer the first scale to the other end of the second ; measure the distance between them again ; and so on. (2). To Estimate the Fraction of a Scale over. This may be done by reading through the microscope the division and fi^ction of a division of the scale corre- sponding to the scratch on the second plug. This gives the length of a portion of the scale as a fraction of the true length which is found in (3). (3). To Determine the true Length of the Ivory Scales. This operation requires two reading microscopes. Focus these two, one on each extreme division of the scales to be measured, taking care that the same edge of the scratch is used as before. Then remove the scale, introduce a standard whose graduation can be assumed to be accurate^ Ch. IV. § 5.] Measurement of the Simple Quantities, ' 75 or whose true length is known, and read by means of the micrometer the exact length, through which the microscopes have to be moved in order that their cross-wires may co- incide with two graduations on the standard the distance between which is known accurately.^ The lengths of all the separate parts of the line between die marks, which together make up the whole distance to be measured have thus been expressed in terms of the standard or of the graduations of the micrometer screw. These latter may be assumed to be accurate, for they are only used to measure distances which are themselves small firactions of the whole length measured (see p. 41). All the data necessary to express the whole length in terms of the standard have thus been obtained. Experiment, — Measure by means of the two given scales and the microscope the distance between the two given points. Enter the results thus : — Distance from the mark on first plug to the end graduation of Scale A 0*1552 in. Distance between end graduations of Scales AandB(i) 0*1015 „ n ». » (2) 0*0683 „ n n n (3) OO572 „ n (4) o'"63 „ ,1 n (5) 0*1184 ,, Total of intervals «... '6269 in. Reading of Scale B at the mark on the second plug . 10*631 „ True length of Scale A . • . . . 12*012,, n n B 11-993 „ Total distance between the marks - 3 X 12*012 + 2 X 1 1*993 + 10*631 + 0*6269 -71*280 in. Observations of similar character will enable us to compare together two scales, such as a metre and a yard. For this purpose two travelling microscopes are required. The slides of the two are mounted on a board so as to be ■ For less accurate measurements the length of the scales may alto be detennined by the use of the beam -compass, § 2. 76 Practical Physics. [Ch. IV. f $. parallel and in the same straight line, the distance between the two being about a yard. Each slide is furnished with a scale of millimetres, and verniers reading to one-tenth of a millimetre are attached to the microscopes. Cross-wires are fixed in the eye-piece of the microscopes. Place the yard-measure on the board parallel to the slides, and focus each microscope on marks on the measure. Set the cross-wire so as to bisect the broad image of a division of the measure, the cross-wire being parallel to the division. Do not observe the actual end of the measure — it is difficult to focus this satisfactorily — but choose some division near the end, say, one inch from the end in each case. To determine which division is chosen, move a piece of paper on the scale until its edge appears just to coincide with the cross-wires, and then note the division by looking at the scale directly. The distance between the cross-wires of the microscopes is now known in terms of the divisions of the measure. Let us suppose that this distance is 34 inches. Read the scale and vernier attached to each microscope ; let the readings be a and ^, a being that of the left-hand microscope, and suppose the scales read from left to right Let / be the distance between the cross- wires of the two microscopes where the scale reading of each is zero. Then 34 = / -|- ^ - a, /. / = 34 in. + tf — ^. Now remove the yard-measure and replace it by the metre scale. Set the cross-wires, as before, on two suitable divisions. This should be done without altering the focus of the microscopes ; if the scale when placed in position is not distinct, it can be raised and supported by wedges of wood of proper thickness. Determine as before the divisions on which the cross-wires are set. Suppose them to be 860 millimetres apart, and let the readings of the microscope scales be a* and b'\ then, as before, / = 860 mm. + a'— b\ .*. 34 inches = 860 mm. -h a'- a — (^ * ^) Ch. IV. § 5. J Measurement of the Simple Quantities. 77 By this experiment we determine the number of millimetres in an inch, assuming the scale of the microscopes and the metre scale to be accurate. Test the accuracy of the slide- scale by comparing it with the metre scale, and then express (i) the yard, (2) the foot, (3) the inch in millimetres, com- paring your results with the recognised values. Enter the results thus : — Distance between selected marks on the yard scale . 34 inches First reading of left-hand microscope , , .8*2 mm. „ „ right-hand microscope . : . 9*9 Distance between marks on metre scale • • . 860 Second reading of left-hand microscope • . .12*5 „ „ right-hand microscope . , 10*4 34 inches - 860 + (12*5 - 8*2) - (io'4 - 9*9) -860 + 3-8 -8638 mm. It A. The Kathetometer Microscope. This is an instrument devised by Prof. Quincke for measuring with great accuracy small vertical heights. A metal stand (a, fig. i) car- ries a microscope m, rest- ing horizontally in two Y- shaped supports ; the under side of the metal stand is cemented to a piece of flat glass. The microscope has a fine micrometer scale in its eye-piece. Resting on three levelling-screws is a small table b. The upper surfiure of this table is flat glass, on which the stand a rests. The stand can be easily moved about into any position, there being very little friction between the two glass sur- 78 Practical Physics. (Ch. IV. ? 5. faces. By dusting lycopodium over the table the adjustment is facilitated. In using the instrument the glass table is first levelled by the aid of the screws and a spirit-level \ the micrometer scale is then set vertical, and the value of a division determined ; to do this a finely divided scale is required. This scale is set vertical ; for this purpose it is convenient to have it attached to a small levelling-table, with a circular level, in such a way that when the level is set the scale is vertical. The microscope b focussed on the scale, and the readings of the micrometer divisions corresponding to the consecutive divisions of the scale are taken ; from these the value of one micrometer division is foimd. The instrument may now be used to determine small differences in vertical height ; if the two marks, the height-difference of which is required, are so placed that they can be brought into the field of view of the microscope simultaneously, the difference of their heights can be read off directly on the scale. When this is not the case, by moving the microscope, bring one mark into focus, and read off its position on the scale ; then, without altering the position of the microscope in the Ys, slide the stand a over the horizontal glass plate until the second mark is in focus, and read its position on the scale. The height of the axis of the microscope above the glass plate and the inclination of the axis to the horizon remain unaltered by this motion, and thus the difference between the two readings gives the difference of height between the two marks. 6. The Kathetometer, This instrument consists of a vertical beam carrying a scale. Along the scale there slides a brass piece, support- ing a telescope, the axis of which can be adjusted so as to be horizontal. The brass slide is fitted with a vernier Ch.iv. f6.i Measurement of tkt Simple QuantitUs, 79 which reads fractions of the divisions of the scale, thus determining the position of the telescope. The kathetometer is used to measure the difference in height between two points. To accomplish this, a level fitted so as to be at right angles to the scale is pennanently attached to the instru- ment, and the scale is placed vertical by means of levelling screws on which the instrument rests. Let us suppose the instrument to be in adjustment, and let p, Q be the two points, the vertical distance between which is required. The telescope of the instrument has, as usual, cross-wires in the eye-piece. Focus the telescope on the mark p, and adjust it until the image of p coincides with the horizontal cross-wire. Then read the scale and vernier. Let the reading be 72115 cm. Kaise the telescope until q comes into the field, and ad- just again till the image of Q fis. }. coincides with the cross- wire; let the reading be 33*375 cm. The difference in level be- tween 7 and Q is 72-135— 33-375, or 38-850 cm. The adjustments are :— (i) To levd the instrument so that the icale ia vertical in all positions. (3) To adjust the telescope so that its axis is horizontal (3) To bring the cross-wire in the focal plane of the telescope into coincidence with the image of the mark which is being ob- served. {i)Thescalemust be vertical, because we use the instru- ment to measure the vertical height between two points. The scale and level attached to it (fig. 5) can be turned 8o Practical Physics. [Qh. iv. | d round an axis which is vertical when properly adjusted, carrying the telescope with them, and can be clamped in any position by means of a screw. (d) To test the Accuracy of the Setting of the Scale4evd and to set the Axis of Rotation verticaL If the scale-level is properly set it is perpendicular to the axis of rotation ; to ascertain whether or not this is so, turn the scale until its level is parallel to the line joining two of the foot-screws and clamp it; adjust these screvrs until the bubble of the level is in the middle. Unclamp, and turn the scale round through i8o^ If the bubble is still in the middle of the level, it follows that this is at right an^es to the axis of rotation ; if the bubble has moved, then the level and the axis of rotation are not at right angles. We may make them so by adjusting the screws which fix the level to the instrument until the rotation through i8o** produces no change, or, without adjusting the level, we may proceed to set the axis of rotation vertical if, instead ci adjusting the levelling screws of the instrument until the bubble stands in the centre of the tube, we adjust them until the bubble does not move relatively to the tube when the instrument is turned through i8o^ This having been secured by the action of two of the screws, turn the scale until the level is at right angles to its former position and clamp. Adjust now in the same manner as before, using only the third screw. It follows then that the bubble will remain unaltered in position for all positions of the instrument, and that the axis about which it turns is verticaL If the scale of the instrument were parallel to the axis, it, too, would be vertical, and the instrument would be in adjustment (b) To set the Scale vertical. To do this there is provided a metallic bracket-piece One arm of this carries a level, while the other b a flat surface at right angles to the axis of the level, so that when Ch. IV. §6.] Measurement of the Simple Quantities. 8i the level is horizontal this surface is truly vertical The adjustment can be tested in the following manner. The level can rotate about its axis, and is weighted so that the same part of the tube remains uppermost as the bracket is. rotated about the axis of the level Place then the flat face of the bracket with the level uppermost against a nearly vertical plane surface ; notice the position of the bubble. Then reverse it so that the level is lowest, and read the posi- tion of the bubble again. If it has not changed the level is truly set, if any displacement has taken place it is not so. The scale of the instrument can be adjusted relatively to the axis of rotation and fixed by screws. Press the flat surface of the bracket- piece against the &ce of the scale. If the scale be vertical, the bubble of the level on the bracket-piece will occupy the middle of its tube. Should it not do so, the scale must be adjusted until the bubble comes to the central positioa We are thus sure that the scale is vertical For ordinary use, with a good instrument, this last ad- justment may generally be taken as made. Now turn the telescope and, if necessary, raise or lower it until the object to be observed is nearly in the middle of the field of view. (2) It is necessary that the axis of the telescope should be always inclined to the scale at the same angle, for if, when viewing a second point q, the angle between the axis and the scale has changed from what it was in viewing p, it is clear that the distance through which the telescope has been displaced will not be the vertical distance between p and Q. If, however, the two positions of the axis be parallel, the difference of the scale readings will give us the distance we require. Now the scale itself is vertical The safest method, therefore, of securing that the axis of the telescope shall be always inclined at the same angle to the scale is to adjust 82 Practical Physics. [Ch. IV. s 6. the telescope so that its axis shall be horizontal The method of doing this will be different for different instru- ments. We shall describe that for the one at the Cavendish ^Laboratory in full detail ; the plan to be adopted for other instruments will be some modification of this. Fig. 6. In this instrument (fig. 6) a level l m is attached to the telescope t t'. The telescope rests in a frame v v'. The lower side of this frame is beveUed slightly at n ; the two surfaces v n, v' n being flat, but inclined to each other at an angle not far from i8o°. Ch. IV. $ 6.] Measurement of the Simple Quantities. 83 This under side rests at n on a flat surface c d, which is part of the sliding-piece c d, to which the vernier v v' is fixed. A screw passes through the piece v y' at n, being fixed into c D. The hole in the piece y y^ is large and somewhat conical, so that the telescope and its support can be turned about N, sometimes to bring n y into contact with c n, sometimes to bring n y' into contact with n d. Fitted into c d and passing freely through a hole in n y' is a screw q ; p is another screw fitted into c d, which bears against n y^. Hidden by p and therefore not shown in the figure is a third screw just like p, also fitted into c d, and bearing against n y*. The screws n, p, and Q can all be turned by means of a tommy passed through the holes in their heads. When p and Q are both screwed home, the level and telescope are rigidly attached to the sliding- piece c D. Release somewhat the screw q. If now we raise the two screws p, we raise the eye-piece end of the telescope, and the level-bubble moves towards that end. If we lower the screws p, we lower the eye-piece end, and the bubble moves in the opposite direction. Thus the telescope can be levelled by adjusting the screws p. Suppose the bubble is in the centre of the level. Screw down the screw q. This will hold the telescope fixed in the horizontal position. If we screw q too firmly down, we shall force the piece N y' into closer contact with the screws p, and lower the eye-piece end. It will be better then to adjust the screw p so that the bubble is rather too near that end of the tube. Then screw down q until it just comes to the middle of the tube, and the telescope is level. {3) To bring the image of the object viewed to coincide with the cross-wires. The piece c d slides freely up and down the scale, eff's' is another piece of brass which also slides up and dowa G 2 84 Practical Physics. [Ch. IV. § 6. H is a screw by means of which e f' can be clamped &st to the scale. A screw R r' passes vertically upwards through E f' and rests against the under side of a steel pin g fixed in c D. Fixed to ef' and pressing do¥mwards on the pin g so as to keep it in contact with the screw r r' b a steel spring s s'. By tinning the screw r r', after clamping h, a small motion up or down can be given to the sliding piece c d and telescope. Now loosen the screw H and raise or lower the two pieces c d, e f' together by hand, until the object viewed is brought nearly into the middle of the field of view. Then clamp E F^ by the screw h. Notice carefully if this operation has altered the level of the telescope ; if it has, the levelling must be done agaia By means of the screw r r' raise or lower the telescope as may be needed tmtil the image is brought into coincidence with the cross- wire. Note again if the bubble of the level is in its right position, and if so read the scale and vernier. It may happen that turning the screw r r' b sufficient to change the level of the telescope. In order that the g^de c D may move easily along the scale, a certain amount of play must be left, and the friction between r' and the pin is sometimes sufficient to cause this play to upset the level adjustment The instrument is on this account a trouble- some one to use. The only course we can adopt is to level and then adjust R r' till the telescope is in the right position, levelling again if the last operation has rendered it necessary. This alteration of level will produce a small change in the position of the line of collimation of the telescope rela- tively to the vernier, and thus introduce an error, unless the axis round which the telescope turns is perpendicular both to the line of collimation and to the scale. If, however, the axis is only slightly below the line of collimation and the change of level small, the error will be very small indeed and may safely be neglected. It is clear that the error produced by an error in levelling Ch IV. § 7. J Measurement of the Simple Quantities. 85 will be proportional to the distance between the instrument and the object whose height is being measured. We should therefore bring the instrument as dose to the object as is possible^ Experiment, — ^Adjust the kathetometer, and compare by means of it a length of 20 cm. of the given rule with Uie scale of the instrument Hang the rule up at a suitable distance from the katheto- meter, and measure the distance between divisions 5 cm. and 25 cm. The reading of the kathetometer scale in each position must be taken three times at least, the telescope being displaced by means of the screw R r' between successive observations. Enter results as below : — KiUh. reading, ttpper mark Kath. reading, lower fnar!i 25315 45325 25"305 45*335 25*320 45*330 Mean 253133 45-330 Difference 20-0167 Mean error of scale between divisions 5 and 25, -0167 cm. MEASUREMENT OF AREAS. 7. Simpler Hethods of measuring Areas of Plane Figures. There are four general methods of measuring a plane area: — (d) If the geometrical figure of the boundary be known, the area can be calculated from its linear dimensions— eg. if the boundary be a circle radius r. Area = ir r^ where it = 3* 142. A table of areas which can be found by this method is given in Lupton's Tables, p. 7. The areas of composite figures consisting of triangles and circles, or parts of circles, may be determined by addition of the calculated areas of all the separate parts. 86 Practical Physics. [Ch. iv. % 7. In case two lengths have to be measured whose product determines an area, they must both be expressed in the same unit, and their product gives the area expressed in terms of the square of that unit (^) If the curve bounding the area can be transferred to paper divided into known small sections, eg. square milli- metres, the area can be approximately determined by count- ing up the number of such small areas included in the bounding curve. This somewhat tedious operation is facili- tated by the usual grouping of the millimetre lines in tens, every tenth line being thicker. In case the curve cuts a square millimetre in two, the amount must be estimated ; but it will be generally sufficient if portions greater than a half be reckoned a whole square millimetre and less than a half zera (r) By transferring the curve of the boundary to a sheet of paper or metal of uniform thickness and cutting it out, and cutting out a square of the same metal of known length of side, say 2 inches, and weighing these two pieces of metal The ratio of their weights is the ratio of the areas of the two pieces of metal The one area is known and the other may therefore be determined. {d) By the planimeter. A pointer is made to travel round the boundary, and the area is read off directly on the graduated rim of a wheel For the theory of this instrument see Williamson's In- tegral Calculus (§ 149). Practical instructions are issued by the makers. Experiment — Draw a circle of 2 in. radius. Calculate or determine its area in all four ways, and compare the results. Enter results thus : — Method a b c d 12566 sq. in. 12*555 sq. in. 12-582 sq. in. 12-573 sq. in. Ch. IV. § 7.] Measurement of the Simple Quantities, 87 Orthogonal Projection, Suppose that through all points of the boundary of an area, % lines are drawn perpendicular to a given plane, the feet of these lines will trace out a curve in the plane ; this curve is said to be the orthogonal projection of the boundary of the given area, and the area bounded by the curve is the orthogonal projection of s. It is easy to see that in orthogonal projection parallel straight lines are projected into parallel straight lines, and the ratio of their lengths is unaltered ; and also that the orthogonal projection of a finite straight line on a plane is equal in length to the length of the projected line multiplied by the cosine of its inclination to the straight line or plane. If s be an area cut out of a sheet of metal or cardboard, the form of its orthogonal projection can be obtained thus : — Place a piece of paper on a horizontal drawing-board, and secure the area s in the required position above it Then hold a plumb-line, made of a piece of thin silk, or cotton, and a shot, so that the plummet is just above the paper, while the line is made to touch in succession a number of points on the boundary of s, and mark the cor- responding position of the plummet with a pencil-dot on the paper. If a sufficient number of points are taken, a curve can be drawn through them, and this curve will be the orthogonal projection of the boundary of s. If the original area be plane, and if a be the angle between its plane and that of the projection, we may shew that the area of the projection is s cos a. For let a b (fig. ii) be the line in which the planes of the two areas intersect Let a line p p', drawn perpendicular to a b, cut the boundary of s in p and p', and let q, q' be the projections of p and p'. Then p' p and q' q when produced meet a b in the same point c, and the angle p c q b a. Let p r, drawn (larallel to Q q', meet p' q' in r. Then q q'= p r = p p' cos a. 88 Practical Physics, [Ch. IV. § 7 Now the area s may be considered as made up of a large number of very narrow j)arallelograms with their lengths Fig. n. parallel to p p' and their breadth parallel to a b. Each o£ these will be projected into a corresponding parallelogram of the same breadth, but of length Q q' or p p' cos a. These projected parallelograms make up the projected area s' ; the area of each parallelogram is decreased by projection in the ratio of cos a to unity. Thus the whole area s projects into an area s cos a. The projection of a circle is a curve called an ellipse. Many of the most important geometrical properties of the ellipse can be very simply deduced by the method of pro- jection from the corresponding properties of a circle (see Clifford's * Elements of Dynamic,' chap. i.). Experiment — Cut out a circle of 3 inches radius. Fix it at an angle of between 30^ and 60^ to the horizon, and project it on to a piece of squared paper. Find the angle a between the area and the horizon, and sliew that the area of the projection - area of circle x cos a. Measure also the maximum and minimum semi-diameters, a and ^, of the ellipse, and shew that the area ^n ab. Ch. IV. §8.] Measurement of the Simple Quantities, 89 8. Determination of fhe Area of the Cross-Section of a Cylindrical Tube. — Calibration of a Tube. The area of the cross-section of a narrow tube is best determined indirectly from a measurement of the volume of mercury contained in a known length of the tube. The principle of the method is given in Section 9. The tube should first be ground smooth at each end by rubbing on a stone with emery-powder and water, and then very carefully cleaned, first with nitric acid, then with distilled water, then with caustic potash, and finally rinsed with distilled water, and very carefully dried by passing air through it, which has been fi-eed from dust by passing through a plug of cotton- wool and dried by chloride of calcium tubes.^ If any trace of moisture remain in the tube, it is very difficult to get all the mercury to run out of it after it has been filled. The tube is then to be filled mihpure^ mercury ; this is best done by immersing it in a trough of mercury of the necessary length. [A deep groove about half an inch broad cut in a wooden beam makes a very serviceable trough for the purpose.] When the tube is quite full, close the ends with the forefinger of each hand, and after the small globules of mercury adhering to the tube have been brushed off, allow the merciuy to run into a small beaker, or other con- venient vessel, and weigh it. Let the weight of the mercury be w. Measure the length of the tube by the calipers or beam-compass, and let its length be /. Look out in the table (33) the density of mercury for the temperature (which may be taken to be that of the mercury in the trough), and ' For this and a great variety of similar purposes an aspirating pump attached to the water-supply of the laboratory is very con- venient. The different liquids may be drawn up the tube by means of an air-syringe. • A supply of pure mercury may be maintained very conveniently by distillation under very low pressure in an apparatus designed l^ Wcinhold (see Carl's Rep. voL 15, and PhiL Mag, Jan. 1884). go Practical Physics. [Ch. IV. § 8. let this be p. Then the volume v of the mercury is given by the equation v=!r. and this volume is equal to the product of the area a of the cross-section and the length of the tube. Hence V ttr If the length be measured in centimetres and the weight in grammes, the density being expressed in terms of grammes per cc., the area will be given in sq. cm. The length of the mercury column is not exactly the length of the tube, in consequence of the fingers closing the tube pressing slightly into it, but the error due to this cause is very small indeed. This gives the mean area of the cross-section, and we may often wish to determine whether or not the area of the section is uniform throughout the length. To do this, care- fully clean and dry the tube as before, and, by partly im- mersing in the trough, introduce a thread of mercury of any convenient length, say about 5 centimetres long. Place the tube along a millimetre scale, and fix it horizontally so that the tube can be seen in a telescope placed about six or eight feet oflf. By slightly inclining the tube and scale, adjust the thread so that one end of it is as close as possible to the end of the tube, and read its length in the telescope. Displace the thread through 5 cm. and read its length again ; and so on, until the thread has travelled the whole length of the tube, taking care that no globules of mercury are left behind. Let /i, /j, /s . . . . be the successive lengths of the thread Then run out the mercury into a beaker, and weigh as before. Let the weight be w^ and the density of the mercury be p. Ch. rv. 1 8.] Measurement of the Simple Quantities. 91 Then the mean sectional areas of the different portions of the tube are www - — 7"> — r> — r» • • • • ^^ p/, p/, p/. The mean of all these values of the area should give the mean value of the area as determined above. The accu- racy of the measurements may thus be tested. On a piece of millimetre sectional paper of the same length as the tube mark along one line the different points which correspond to the middle points of the thread in its difieient positions, and along the perpendicular lines through these pomts mark off lengths representing the correspond- ing areas of the section, using a scale large enough to shew clearly the variations of area at different parts of the length. Join these points by straight lines. Then, the ordinates of the curve to whidi these straight lines approximate give the cross-section of the tube at any point of its length. Experimenf, — Calibrate, and determine the mean area of the given tube. Enter the result thus : — [The results of the calibration are completely expressed by the diagram.] Length of tube . . . 25*31 cm. Weight of beaker • . . 10*361 gm. Weight of beaker and mercury . 11786 gm. Weight of mercury . . 1*425 gm. Temperature of mercury . •14® C. Density of mercury (table 33) 13*56 Mean area of section- Lli^ — . sq. cm. 25*31 X 13-56 -0*415 sq. mm. Mean of the five determinations for calibration 0*409 sq. mm. 92 Practical Physics. [Ch. IV. § 9. MEASUREMENT OF VOLUMES. The volumes of some bodies of known shape may be de- termined by direct calculation from their linear dimensions ; one instance of this has been given in the experiment with the calipers. A Table giving the relations between the volume and linear dimensions in those cases which are likely to occur most frequently will be found in Lupton's Tables, p. 7. 9. Determination of Volumes by Weighing. Volumes are, however, generally determined from a knowledge of the mass of the body and the density of the material of which it is composed. Defining ' density ' as the mass of the unit of volume of a substance, the relation between the mass, volume and density of a body is ex- pressed by the equation M=Vf>, where m is its mass, v its volume, and p its density. The mass is determined by means of the balance (see p. 123), and the density, which is different at different temperatures, by one or other of the methods described below (see pp. 139-143). The densities of certain substances of definitely known composition, such as distilled water and mercury, have been very accurately de- termined, and are given in the tables (Nos. 33, 33), and need not therefore be determined afresh on every special occa- sion. Thus, if we wish, for instance, to measure the volume of the interior of a vessel, it is sufficient to detemune the amount and the temperature of the water or mercury which exactly fills it This amount may be determined by weigh- ing ^ the vessel full and empty, or if the vessel be so large that this is not practicable, fill it with water, and run the water off in successive portions into a previously counterpoised flask, holding about a litre, and weigh the flask thus filled. Care must be taken to dry the flask between the successive fillings ; this may be rapidly and easily done by using a hot clean cloth. The capacity of vessels of very considerable » For exart work the weighings must be corrected for the buoyancy of the air. See p. 12a Ch. IV. § 10.] Mettsurenient of the Simple Quantities. 93 size may be detennined in this way with very great accuracy. All tke specific gravity experiments detailed below involve the measurement of a volume by this method. Experiment, — Determine the volume of the given vessel. Enter results thus : — Weight of water < I • I » . , . --v^ ^Mvr'c;>:rYy Filling I • . 1001*2 gms. 2 . . 9987 w 3 • • 1002-3 ^ '^ ^/r -pp. y,j^ 4 . . 999*2 „ 5 . . 798-1 I, Total weight • 4799' 5 gms. Tecflperature of water Volume . . 48035 CO. ■ in vessel, 15**. 10. Testing the Accuracy of the Graduation of a Burette. Suppose the burette to contain 100 cc ; we will suppose also that it is required to test the capacity of each fifth of the whole. The most accurate method of reading the burette is by means of z. floaty which consists of a short tube of glass loaded at one end so as just to float vertically in the liquid in the burette ; round the middle of the float a line is drawn, and the change of the level of the liquid is determined by reading the position of this line on the graduations of the burette. The method of testing is then as follows :— Fill the burette with water, and read the position of the line on the float Carefully dry and weigh a beaker, and then run into it from the burette about ^th of the whole contents ; read the position of the float again, and weigh the amount of water run out into the beaker. Let the number of scale divisions of the burette be 20*2 and the weight in grammes 20 '119. Read the temperature of the water ; then, knowing the density of water at that temperature (from table 32), and that i gramme of water at 4'' C. occupies i cc. 94 Practical Physics. [Ch. iv. § lou we can detennine the actual volume of the water correspond- ing to the 3o*2 cc as indicated by the burette, and hence determine the error of the burette. Proceeding in this way for each ^th of the whole volume, form a table of cor- rections. Experiment — Form a table of corrections for the given burette. Enter results thus : — Burette readings Error o - 5 cc. . • • — •007 CC 5 -lo w • • • -•020 „ lO -15 n . • • -•on „ 15 -20 „ . • • •000 „ 20 -25 „ . • . -•036 „ MEASUREMENT OF ANGLES. The angle between two straight lines drawn on a sheet of paper may be roughly measured by means of a protractor, a circle or semi-circle with its rim divided into degrees. Its centre is marked, and can therefore be placed so as to coin- cide with the point of intersection of the two straight lines ; the angle between them can then be read off on the gradua- tions along the rim of the protractor. An analogous method of measuring angles is employed in the case of a compass- needle such as that required for § 69. Angles traced on a diagram may be determined by measuring lines from which one or other of the trigonometrical ratios can be calculated (see Chap. V.»). The more accurate methods of measuring angles depend on optical principles, and their consideration is accordingly deferred until the use of the optical instruments is explained (see §§62, 71). MEASUREMENT OF SOLID ANGLES. The angle which a plane curve joining any two points subtends at a third point o in the plane of the curve, as given by its 'circular measure,' may be found thus : — * Ch. IV. § la] Measurement of the Simple Quantities. 95 Let A B be the curve. Join o a, o b, and with o as centre and any radius describe a circle a' b', cutting o A, o B in a' and b'. The ratio of the arc a' b' to the radius o a' is the same for all values of the radius o a', and is the mea- sure of the angle a o b in ' radians '; if the radius o a' be unity, then the arc a'b' measures the angle. The circular measure of an angle is the number of units of length in the arc of a circle of unit radius sub- tended by the angle. A corresponding method is employed to measure the * solid angle' subtended at a point by a surface. Let o be the point, a b c the surface (fig. iv). Witho Fig. in as centre and any radius de- scribe a sphere, and consider a line passing through o which °/ moves so as to trace out the boundary of the area a b c. It will thus describe a cone cut- ting the sphere in a closed curve h! b' d^ and we can shew that the ratio of thiS area to the square of the radius oa' is the same for all values of the radius. This ratio is adopted as defining the measure of the solid angle at o. If we take a sphere of unit radius, the ratio becomes the measure of the area a' b' d, and we thus find that the solid angle subtended by an area at a point is measured by the number of units of area intercepted from a sphere of unit radius by a cone with the given point as vertex and the given area as base. If the area as seen from the given point appears circular in form, the cone is a right circular cone and the boundary h! ^ d on the sphere is a circle Let o l p (fig. v) 96 Practical Physics. [Ch. IV. f la be the axis of this cone, and let o a', the radius to any point on the circle, be inclined at an angle a to o P. De- scribe a cylinder with its axis parallel to o p touching the sphere. The circle a Bclies in a plane perpendicular to o p. Let this circle cut c the cylinder in the circle D B, and let a plane touch- ing the unit sphere in L cut the cylinder in fg. Then by an application of the method of projection it may be shewn that the area of the belt of the cylinder between d e and f g is equal to the corresponding area l a' b' c' on the sphere, and this last measures the required solid angle at o. Let M be the centre of the circle a' b' c'. The solid angle = area of belt f d e G = 27r D M . L M ; LM = LO — OM = LO — OA' COS a = I — COS n ; for L o = o a' = I, the sphere being of unit radius. Also D M = I. /. Solid angle = 27r (i — cos a). This expression, of course, only holds when the solid angle in question is that of a right circular cone. It is clear from the above that a * solid angle ' is not an angle at all, but is only so named from analogy, being related to a sphere of unit radius in a manner similar to the relation between the circular measure of an angle and the circle of unit radius. measurements of time. The time-measurements most frequently required in practice are determinations of the period of vibration of a needle. To obtain an accurate result some practice in the use of the * eye and ear method ' is required. The experi- Cm. rv. § II.] Measurement oj the Simple Quantities. 97 ment which follows (§11) will serve to illustrate the method and -also to call attention to the fact that for accurate work any clock or watch requires careful ' rating,' Le. comparison of its rate of going with some timekeeper, by which the times can be referred to the ultimate standard — the mean solar day. The final reference requires astronomical obser- vations. Different methods of time measurement will be found in §§ 21 and 28. The * method of coincidences ' is briefly discussed in § 20. II. Bating a Watch by means of a Seconds-Clock. The problem consists in determining, within a fraction of a second, the time indicated by the watch at the t^o instants denoted by two beats of the clock with a known interval between them. It will be noticed that the seconds- finger of the clock remains stationary during the greater part of each second, and then rather suddenly moves on to the next point of its dial. Our object is to determine to a fraction of a second th^ time at whidfit just completes one of its jomneys. To do this we must employ both the eye and ear, as it is impossible to read both the clock and watch at the same instant of time. As the watch beats more rapidly than the dock, the plan to be adopted is to watch the latter, and listening to the beating of the former, count along with it until it can be read. Thus, listening to the ticking of the watch and looking only at the clock, note the exact instant at which the clock seconds-finger makes a particular beat, say at the completion of one minute, and count along with the watch-ticks from that instant, beginning o, i, 2, 3, 4, . . and so on, until you have time to look down and identify the position of the second-hand of the watch, say at the instant when you are counting 21. Then we know that this time is 31 ticks of the watch after the event (the clock-beat) whose H 98 Practical Physics. [Ch. IV. 5 i». time we wished to register ; hence, if the watch ticks 4 times a second, that event occurred at V seconds before we took the time on the watch. We can thus compare to within \ sec. the time as indicated by the clock and the watch, and if this process be repeated after the lapse of half an hour, the time indicated by the watch can be again compared, and the amount gained or lost during the half-hour determined. It will require a little practice to be able to count along with the watch. During the interval we may find the number of ticks per second of the watch. To do this we must count the number of ticks during a minute as indicated on the clock. There being 4 or 5 ticks per second, this will be a difficult operation if we simply count along the whole way; it is there- fore better to count along in groups of either two or four, which can generally be recognised, and mark down a stroke on a sheet of paper for every group completed ; then at the end of the minute count up the number of strokes ; we can thus by multiplying, by 2 or 4 as the case may be, obtain the number of watch-ticks in the minute, and hence arrive at the number per second. Experiment — Determine the number of beats per second made by the watch, and the rate at which it is losing or g^ning. Enter results thus : — No. of watch-ticks per minute, 100 groups of 3 each. No. of ticks per second, 5. hr. m. s* Clock-reading 11 38 3 Estimated watch-reading, 11 hr. 34 m. and 10 ticks » 11 34 2 Difference - 41 Clock-reading. • . . . . . 12 8 3 Estimated watch-reading, 12 hr. 4 m. and 6 ticks « 12 4 ra Difference . . . 418 Losing rate of watch, 1*6 sec per houc 99 CHAPTER V. MEASUREMENT OP MASS AND DETERMINATION OP SPECIFIC GRAVITIES. 12. The Balance. General Considerations. The balance, as is well known, consists of a metal beam, supported so as to be free to turn in a vertical plane about an axis perpendicular to its length and vertically above its centre of gravity. At the extremities of this beam, pans are sus- pended in such a manner that they turn freely about axes, passing through the extremities of the beam, and parallel to its axis of rotatioa The axes of rotation are formed by agate knife-edges bearing on agate plates. The beam is provided with three agate edges; the middle one, edge down- wards, supporting the beam when it is placed upon the plates which are fixed to the pillar of the balance, and those at the extremities, edge upwards ; on these are supported the agate plates to which the pans are attached. The effect of hanging the pans firom these edges is that wherever in the scale pan the weights be placed, the vertical force which keeps them in equilibrium must pass through the knife-edge above, and so the effect upon the balance is independent of the position of the weights and the same as if the whole weight of the scale pan and included masses were collected at some point in the knife-edge from which the pan is suspended. In order to define the position of the beam of the balance^ a long metal pointer is fixed to it, its length being perpen- dicular to the line joining the extreme knife-edges. A small scale is fixed to the pillar of the balance, and the motion of the beam is observed by noting the motion of the pointer along this scale. When the balance is in good adjustment, the scale should be in such a position that the pointer is lOO Practical Physics, [Ch. V. % la. opposite the middle division when the scale-beam is hori- zontal The only method at our disposal for altering the relative position of the scale and pointer is by means of the levelling screws attached to the case. Levels should be placed in the case by the instrument-maker, which should shew level when the scale is in its proper position. In the investigation below we shall suppose the zero position of the balance to be that which is defined by the pointer being opposite the middle point of its scale, whether the scale is in its proper position, and the pointer properly placed or not The other conditions which must be satisfied if the balance is in perfect adjustment are : — (i) The arms must be of equal length. (2) The scale pans must be of equd weight (3) The centre of gravity of the beam must be vertically uhder the axis of rotation when the beam is in its zero position. This can always be ensured by removing the scale pans altogether, and by turning the small flag of metal attached to the top of the beam until the latter comes to rest with the pointer opposite the middle of its scale. Then it is obvious from the equilibrium that the centre of gravity is vertically under the axis of support On the Sensitiveness of a Balance, Let us suppose that this third condition is satisfied, and that the points a, c, b (fig. 7) represent the points in which pj^ the three knife-edges cut a vertical plane at B right angles to their \qp edges, and let c a, cb make angles a, o! with a horizontal line through c [If the balance is in perfect adjustment a=a'.] We may call the lengths c a, c b the lengths of the arms Ch. V. § 12.] Measurement of Mass, fOl of the balance, and represent them by r, l respectively. Let the masses of the scale pans, the weights of which act ver- tically downward through a and b respectively, be p and Q. Let G, the centre of gravity of the beam, be at a distance h^ vertically under c, and let the mass of the beam be k. If the balance be in adjustment, R is equal to l, and p to q. Now let us suppose that a mass w is placed in the scale pan p, and a mass w+xinQy and that in consequence the beam takes up a new position of equilibrium, arrived at by turning about c through an angle ^, and denoted by b' c a', and let the new position of the centre of gravity of die beam be q\ Then if we draw the vertical lines b' m, a' n to meet the horizontal through c in m and n, a horizontal line through g' to meet c o in x, and consider the equilibrium of the beam, we have by taking moments about the point c (Q+a/+«) CM = (p+ze/) ON + k.g'x. Now CM = cb' cos (a'— fl) = l(cos a' cos 6 + sin a' sin 6), CN = CA'cos(a + ^)=sR(cosa cosd — sin a sin 6). g' X = c g' sin tf = A sin A Hence w^ get L (q+w-\-x) (cos a' cos ^+sin a' sin 0) =R(p+a') (cos a cos tf— sin a sin tf) + K^ sin A Since 6 is very small, we may write tan 0=^, • B—txTifi— ^' (Q"^^^'*"^) ^^^ g^— r(p+w) cos g . . kA — l(q+«'+^) sin a'— r(p + «') sin a*^ ' This gives us the position in which the balance will rest when the lengths of the arms and masses of the scale pans are known, but not necessarily equal or equally inclined to the horizon; and when a difference x exists between the masses in the scale pans. It is evident that may be expressed in pointer scale divisions when the angle subtended at the axis of rotation by one of these divisions is known. 102 Practical Physics. [Ch. V. § i«. Definition. — The number of scale divisions between the position of equihbrium of the pointer when the masses are equal and its position of equilibrium when there is a given small difference between the masses is called the sensitiveness of the balance for that small difference. Thus, if the pointer stand at loo when the masses are equal and at 67 when there is a difference of *ooi gramme between the masses, the sensitiveness is 33 per milligramme. We have just obtained a formula by which the sensi- tiveness can be expressed in terms of the lengths of the arms, &c Let us now suppose that the balance is in adjustment, Le. L=R, Q=P, a=a' Hence the angle turned through for a given excess weight x increases proportionally with x^ and increases with the length of the arm. Let us consider the denominator of the fraction a little more closely. We see that it is positive or negative ac- cording as Ki^>or <L (2P+22c;+^) sin cu Now it can easily be shewn that the equation K>4=L(2p+2Z«;+:c)sin a leads to the condition that if a: be zero, c is the centre of gravity of the beam and the weights of the scale pans &c. supposed collected at the extremities of the arms. In this case with equal weights w in the scale pans, the balance would be in equilibrium in any position. If K >4 be less than l(2P+ 2££;+;c) sin a, tan « is n^ative, which shews that there is a position of equihbrium with the centre of gravity of the whole, above the axis ; but it is reached by moving the beam in the opposite direction to that Cm. V. § I2.J Measurement of Mass, 103 in which the excess weight tends to move it : it is therefore a position of unstable equilibnum. We need only then discuss the case in which k ^ is > L(2P+2^+^)sinay i.e. when the centre of gravity of the whole is below the axis of rotation. With the extreme knife-edges above the middle one, a is positive and the denominator is evidently diminished, and thus the sensitiveness increased, as the load w increases; but if the balance be so arranged that a=o, which will be the case when the three knife-edges are in the same plane, we have tan ^= — ;, or the sensitiveness is independent of the load ; if the extreme knife-edges be below the mean, so that a b nega- tive^ then the denominator increases with the load w^ and consequently the sensitiveness diminishes. Now the load tends to bend the beam a little ; hence in practice, the knife-edges are so placed that when half the maximum load is in the scale pans, the beam is bent so that all the knife- edges lie in a plane, and the angle a will be positive for loads less than this and negative for greater loads. Hence, in properly made balances, the sensitiveness is very nearly independent of the load, but it increases slightly up to the mean load, and diminishes slightly from the mean to the maximum load. The Adjustment of a Balance, I. Suppose the balance is not known to be in adjust- ment. Any defect may be due to one of the following causes: — (i) The relative position of the beam and pointer and its scale may be wrong. Thb may arise in three ways: (a) the pointer may be wrongly fixed, (fi) the balance may not be level, (y) the pointer when in equilibrium witli the pans unloaded may not point to its zero position. We I04 Practical Physics. [Ch. V. § 12. always weigh by observing the position of the pointer when at rest with the scale pans empty, and then bring its position of equilibrium with the pans loaded back to the same point It is clear that this comes to the same thing as using a pointer not properly adjusted. In all these cases a will not be equal to a' in equation (i). (2) The arms may not be of equal length, i.e. l not equal to r. (3) The scale pans may not be of equal weight We may dispose of the third fault of adjustment first If the scale pans be of equal weight, there can be no change in the position of equilibrium when they are interchanged ; hence the method of testing and correcting suggests itself at once (see p. 10 1). The first two faults are intimately connected with each other, and may be considered together. Let the pointer be at its mean position when there is a weight w m v and v/ •\-x in Q, Of and a/ being weights which are nominally the same, but in which there may be errors of small but un- known amount, Then tf=o /. tan 9=o .'. from (i) (assuming p=q) l(p + w' + ^) cos a'=R(p + w) cos a ... (3) Interchange the weights and suppose now that z«/ in q balances a^'+j', in p, then L (p + w) cos a'= R(p + a/ A^y) cos a , - . (4) And if the pointer stands at zero when the pans are un- loaded, we have L. P cos a'= R. P cos a .... (5) Hence equations (3) and (4) become L (zef'+:!c) cos a'=R uf cos a. L w cos a! =R (a/ +^) cos a. Ch. V. § 12.] Measurement of Mass. 105 Multipljring l' cos V (a/+^)=R* (a/+_y) cos *a . . . (6) Lcosa' /W'^y • • R cos a V vx -irx = I '^-~ approximately (p. 44). It will be seen on reference to the figure that l cos a' and R cos a are the projections of the lengths of the arms on a horizontal plane — Le. the practical lengths of the arms considered with reference to the effect of the forces to turn the beam. If the balance be properly levelled and the pointer straight a=a', and we obtain the ratio of the lengths of the actual arms. We thus see that, if the pointer is at zero when the balance is unloaded, but the balance not properly levelled, the error of the weighing is the same as if the arms were unequal, provided that the weights are adjusted so as to place the pointer in its zero position. The case in which a = — a' and therefore cos a = cos a' will be an im- portant exception to this; for this happens when the three knife-edges are in one plane, a condition which is very nearly satisfied in all delicate balances. Hence with such balances we may get the true weight, although the middle point of the scale may not be the equilibrium position of the pointer, provided we always make this equilibrium position the same with the balance loaded and unloaded. If we wish to find the excess weight of one pan from a knowledge of the position of the pointer and the sen- sitiveness of the balance previously determined, it will htt xo6 Practical Physics, fCn. V. § 12. a more complicated matter to calculate the effect of not levelling. We may proceed thus : Referring to equation (i), putting p = Q we get g^ L (pH-gg/+;g) cos g^— R (P-hzy) cos g ""k i4— L (p+ w+jp) sin g' - R (p+a/) sin a* And since ^=0 when no weights are in the pans, we get L P cos a'=R P cos a. kA— L (w+p+jp) sin g'— R (w+p) sin a Since a and o! are always very small, we may put cos a' SSI and sin o!^:^a!^ and so on, the angles being measured in circular measure (p. 45). ^LX .\ tan ^- KA-L(p+a/+jt)g'-R(p+a/)a Lar. r L(zg/-hP -far) g^-fR («; + ?) a "! Neglecting x and the difference between l and p^ in the bracket, since these quantities are multiplied by g or g', we have t«n fl L^gf L(tt; + p)(g^ + an *^^=oL'^ — o J The error thus introduced is small, unless l(wH-p) Kh is a very large quantity, compared with a, and it well may be so, since h is small and w+p may be many times k ; but g in a well-made balance is generally so small that the effect is practically imperceptible, and if the knife-edges be in a plane, so that g = — a', the correction vanishes. Ch. V. § IJ.J Measurement of Mass. 107 Practical Details of Manipulation. Method of Oscillations. All delicate balances are fitted with a long pointer fixed to the beam, the end of which moves over a scale as the beam turns. The middle point of this scale should be vertically be- low the fulcrum of the beam, and if the balance be in perfect adjustment, when the scale pans are empty and the beam free, the end of the pointer will coincide with the middle division of the scale. This coincidence, however, as we have seen, is not rigorously necessary. To weigh a body we require to determine first at what point of the scale the pointer rests when the pans are empty. We then have to put the body to be weighed in one pah and weights in the other, until the pointer wiU again come to rest opposite to the same division of the scale. The weight of the body is found by adding up the weights in the scale pan. We shall suppose that the weights used are gnunmes, decigrammes, &c The weights in the boxes usually supplied are some of them brass and the others either platinum or aluminium. The brass weights run firom i gramme to 50, 100 or 1000 granmies in diflerent boxes. We may divide the platinum and aluminium weights into three series : — The first includes, -5, -2, -i, -i gramme The second '05, -02, '01, -oi „ The third '005, '002, •001, '001 „ that is, the first series are decigrammes, the second centi- grammes, and the third milligrammes. The weights should never be touched with the fingers ; they should be moved by means of the small metal pliers provided for the purpose. In the larger boxes a brass bar 18 provided for lifting the heavier weights. When the balance is not being used, the beam and the scale pans do not rest on the knife-edges but on independent io8 Practical Physics, [Ch. V. § la. supports provided for them. The balance is thrown into action by means of a key in the front of the balance case. This must always be turned slowly and carefully, so as to avoid any jarring of the knife-edges from which the beam and scale pans hang. When it is necessary to stop the beam from swinging, wait until the pointer is passing over the middle of the scale, and then turn the key and raise the frame till it supports the beam. The key must not be turned, except when the pointer is at the middle of the scale ; for if it be, the sup- porting frame catches one end of the beam before the other, and thus jars the knife-edges. The weights or object to be weighed when in the scale pans must never be touched in any way while the beam is swinging ; thus, when it is required to change the weights, wait until the pointer is passing across the middle point of the scale, turn the key, and fix the beam, then move the weights from the scale pan. In the more delicate balances, which are generally en- closed in glass cases, it will be seen that the length of each arm of the beam is divided into ten parts. Above the beam, and slightly to one side of it, there is a brass rod which can be moved from outside the balance case. This rod cames a small piece of bent wire, which can, by moving the rod, be placed astride the beam. This piece of wire is called a * rider.' The weight of the rider is usually one centigramme. Let A c B, fig. 8, be the beam, c being the fulcrum; the divisions on the arm are reckoned from c Suppose now we place the centigramme rider at division I, that is one-tenth of the length of the arm away from the Fig. 8. fulcrum, it will clearly A require one-tenth of its own weight to be placed ^ in the scale pan sus- pended from B, to balance it. The effect on the balance- Cfu V. 5 12.) Measurement of Mass, 109 beam of the centigramme rider placed at division i, is the same as that of a weight of ^^^ centigramme or i milligramme in the pan at a. By placing the nder at division i, we practically increase the weight in the pan at a by i milli- gramme. Similarly, if we place the rider at some other division, say 7, we practically increase the weight in a by 7 milligrammes. The rider should not be moved without first fixing the balance beam. Thus without opening the balance-case we can make our final adjustments to the weights in the scale pan by moving the rider from outside. The object of the case is to protect the balance from draughts and air currents. Some may even be set up in- side the case by opening it and inserting the warm hand to change the weights ; it is therefore important in delicate work to be able to alter the weight without opening the We proceed now to explain how to determine at what point of the graduated scale the pointer rests when the pans are empty. If the adjustments were quite correct, this would be the middle point of the scale. In general we shall find that the resting-point is somewhere near the middle. We shall suppose for the present that the stand on which the balance rests is level. This should be tested by the spirit-level before beginning a series of weighings, and if an error be found, it should be corrected by moving the screw- feet on which the balance-case rests. We shall find that the balance when once set swinging will continue in motion for a long period. The pointer will oscillate across the scale, and we should have to wait for a very long time for it to come to rest We require some method of determining the resting- point from observations of the oscillations. Let the figure represent the scale, and suppose, reckoning from the left, we call the divisions o, 10, 20, 30. . . . I lo Practical Physics. [Ch. V. § la. A little practice enables us to estimate tenths of these divisions. Watch the pointer as it moves ; it will come for a moment to rest at Pi suppose, and then move back again. Note the Fio. 9. division of the scale, « I jF^ 63, at which this hap- liMlhr*!! In^nlnnl P^ns.' The pointer • aastStMM ttirt MMJMUUMIMIMINIMIMINISOIM SWmgS On paSt thC resting-point, and comes to instantaneous rest again in some position beyond it, as Pj, at 125 suppose. Now if the swings on either side of the resting-point were equal, this would be just half-way between these two divi- sions, that is at 94 ; but the swings gradually decrease, each being less than the preceding. Observe then a third turning point on the same side as the first, P3 suppose, and let its scale reading be 69. Take the mean 66, between 69 and 63. We may assume that this would have been the turning-point on that side at the moment at which it was 135 on the other, had the pointer been swinging in the opposite direction. Take the mean of the 135 and 66, and we have 95 *s as the value of the resting-point Thus, to determine the resting point : — Observe three consecutive turning points, two to die left and one to the right, or vice versA. Take the mean of the two to the left and the mean of this and the one to the right ; this gives the resting-point required. The observations should be put down as below. Turning-points Resting-point Left Right Mean 66 [^^ 125 955 We may, if we wish, observe another turning-point to the right, 120 suppose; then we have another such series. * A small mirror is usually fixed above the scale, the planes of the two being parallel. When making an observation the observer's eye is placed so that the pointer exactly covers its own image formed in the mirror ; any error due to parallax is thus avoided. Ch. V. s I J.] Measurement of Mass, in Proceeding thus we get a set of determinations of the resting-point, the mean of which will give us the true position with great accuracy. Having thus found the resting point with the pans empty, turn the key or lever, and fix the beam ; then put the object to be weighed in one scale pan. Suppose it to be the lei^-hand, for clearness in the description. Then put on some weight, 50 grammes say, and just begin to turn the key to throw the balance into action. Suppose the pointer moves sharply to the left, 50 gms. is too much. Turn the key back, re- move the 50 and put on 20 ; just begin to turn the key ; the pointer moves to the right, 20 is too little. Turn the key back, and add 10 ; the pointer still moves to the right ; add 10 more, it moves to the left ; 40 is too much. Turn the key back, remove the 10 and add 5. Proceed in this way, potting on the weights in the order in which they come, re- moving each weight again if the pointer move sharply to the left, that is, if it be obviously too much, or putting on an additional weight if the pointer move to the right There is no necessity to turn the key to its full extent to decide if a weight be too much or too litde until we get very nearly the right weight ; the first motion of the pointer is sufficient to give the required indication. It saves time in the long run to put on the weights in the order in which they come in the box. Caution. — The beam must always be fixed before a weight is changed. Suppose now we find that with 37*68 grammes the pointer moves to the right, shewing the weight too little, and that with 37-69 the motion is to the left, shewing that it is too much. Close the balance-case, leaving on the lighter weight, 37'68 grammes. Turn the key, and notice if the pointer will swing off the scale or not Suppose it is quite clear that it will, or that the resting-point will be quite at one end near the division 20a Fix the beam, and put on the rider say 112 Practual Physics. (Ch. V. s la. at division 2. This is equivalent to adding '002 gm. to the weights in the scale pan, so that the weight there may now be reckoned as 37 '682 gms. Release the beam, and let it oscillate, and suppose that this time the pointer remains on the scale. Read three turning-points as before. TurniDg-points Resting-point Left Right Meani7o|J^» 98 134 Thus we find that with no weights in the scale pans, the resting-point is 95 '5 — we may call this 96 with sufficient ac- curacy — while, with the object to be weighed in the left pan, and 37*682 grammes in the right, the resting-point is 134. Hence 37*682 gms. is too small, and we require to find what is the exact weight we must add to bring the resting point from 134 to 96, that is, through 38 divisions of the scale. To effect this, move the rider through a few divisions on the beam, say through 5 ; that is, place it at division 7. The effective weight in the scale pan is now 37*687 gms.; observe as before. Turning-points Resting-point Left Right Mean 46 1 ^g 102 74 Ihe addition of "005 gramme has moved the resting- point from 134 to 74 ; that is, through 60 divisions. We have then to determine by simple proportion what weight we must add to the 37*682 in order to move the resting-point through the 38 divisions ; that is, from 134 to 96. The weight required is |^ x *oo5 or "003 1 6 gnu If then we add "003x6 gm. to the 37*682, the resting-point will be 96, the same as when the scale pans were empty. Thus the weight of the body is 37*68516 gms. We have not been working with sufficient accuracy to make the last figure at all certain; we will therefore discard it, and take the weight as 37-6852 grammes (p. 37). Ch. V. § 18.] Measurement of Mass, 113 One or two other points require notice. In each case we have supposed the pointer to swing over firom 60 to 70 divisions ; this is as large a swing as should be allowed We have supposed the resting point, when the balance was unloaded, to lie between those for the two cases in which the load was 37*682 and 37*687 ; the weights should always be adjusted so that the like may be the case. We have supposed that the weight for which we first observe the swing is too small. It is more convenient that this should be so ; it is not absolutely necessary : we might have started from the heavier weight, and then moved the rider so as to reduce the weight in the right-hand pan. We must be careful to make no mistake as to the weights actually in the scale paa It is generally wise for beginners to add them up as they rest on the pan, putting down each separately, grouping those weights together which belong to each separate digit, thus arranging them in groups of grammes, decigrammes, centigrammes, and milligrammes, and then to check the result by means of the vacant places left in the box When the weighing is completed see that the weights are replaced in their proper positions in the box, and that the beam is not left swinging. We shall in future refer to this method of weighing as the ' method of oscillations.' The alteration produced in the position of the resting point for a given small addition to the weights in the pan is called, as we have seen, the sensitiveness of the balance for that addition (p. 102). Thus in our case the resting-point was altered by 60 for an addition of '005 gramme. The sensitiveness, then, is 60/5 or 12 per milligramme. The load in the pans in this case was nearly 38 grammes. We should find by experiment that the sensitiveness depends slightly on the load in the pans. (See p. 102). I 114 Practical Physics, [Ch. V. % fx Experiments, (i) Determine the position of the resting -point four times ivhen the balance is unloaded. (2) Weigh the given body twice. (3) Determine the sensitiveness for loads of 10^ 50, and 100 gms. Enter results thus : — (x) Balance unloaded. Resting-point . • 95*5 958 96-1 954 Mean • (2) Weight of the body, ist weighing . 2nd n Mean 957 37-6852 37-6855 37-68535 (3) Sensitiveness. Weight in right>liaod pan 10 granmies 10005 „ 50 u 50-005 .. 100 H 100*005 „ Resting point p^^^uJSSie 134 86 128 281 70) 129) 76) 9*6 11*6 IO-6 13. Testing the Adjustments of a Balance. The method of weighing which we have described in the preceding section requires the balance to be in perfect adjustment But the only precaution for that purpose to which attention was called in the description was the levelling of the balance case. We previously mentioned, however (p. 100), that the centre of gravity of the beam could be made to be vertically under its axis of rotation by adjusting the metal flag attached to the beam, and we have, moreover, shewn (pp. 104, 106) that the effect upon the weighings of the pointer not being properly placed, or of our rot using the middle point of its scale as the zero, is Cm. V. § 13.] Measurement of Mass. 115 inappreciable. We need consider, therefore, only the adjust- ment to equality of the weights of the scale pans and of the lengths of the arms. The former may, if necessary, be made equal by filing one of them until the necessary equality is attained, while the latter can be adjusted by means of the screws which attach the end knife-edges to die beam. We have, however, said nothing as yet about adjusting the sensitiveness of the balance. A delicate balance is generally provided with a small sphere fixed to the beam vertically above the middle knife-edge, whose height can be altered by means of the vertical screw passing through its centre, by which it is supported. By raising or lowering this sphere, called the inertia bob, we can diminish or increase the value of h in equation (i) (p. 1 01), and thus increase or diminish the sensitiveness of the balance. At the same time the moment of inertia (see p. 190) of the beam about the axis of rotation is correspondingly increased, and with it the time of swing of the pointer. Now a long period of swing involves spending a long time over the weighings, and this is a disadvantage ; it is therefore not advisable to make the sensitiveness so great that the time of swing is inconveniently long. The usual period of swing is about 15 seconds. Lord Rayleigh has, however, recently suggested (Brit Assoc. 1883) that the same accuracy of weighing with considerable saving of time may be secured by loading the pointer of the balance so that the time of swing is about 5 seconds, and using a magnifying glass to read the turning points of the pointer, and thus making up for the diminished sensitiveness by increased accuracy of reading. None of these adjustments should be carried out by any but practised observers with the balance, and not by them except after consultation with those who are responsible for the safe custody of the instrument It is, however, very important for every observer to be able to tell whether or not the balance is in adjustment, and we therefore proceed I 2 zr w w 1 16 Practical Physics. [Ch. V. { 13. to give practical directions for testing in such a manner as to measure tlie errors produced and enable us to allow for them. (i) To determine the Ratio of the Arms of a Balance^ and to find the true Weight of a Body by means of a Balance with unequal Arms. Let A c B be the beam, and let r and l be the lengths of Fig. 10. the arms c b and c a. A I. C R _B Weigh a body, whose true weight is w, in the right-hand scale ' "W i^iPan, and let the ap- parent weight be w,. Then weigh it in the lei^-hand pan, and let the apparent weight be Wj. The weighing must be done as described in the previous section. Then we have WXR=W,XL (0 WjXR=WXL (2) Provided that pxr=qxl, where p and q are the weights of the scale pans — i.e. provided the balance pointer stands at zero with the pans unloaded. In practice this condition must first be satisfied by adding a counterpoise to one of the pans. Multiplying (i) by (2) w, X R*=Wi X L*, R* W. R /w7 / . or -5=—^ ~ = a/ —••••• (3) L* W, L V W, ^^' Dividing (i) by (2) w__w, Wi w" W*=sW, xws w=\/w,xw, • • • (4) When Wi and Wj are nearly the same, we may put Ch. V. f 13.] Measurement of Mass, 117 ftw V w, w,, i(wi+w,), since the error depends on { ^/^\ — >/wj} *, and this quantity is very small. (See p. 45.) Thus, if w„ Wj be the apparent weights of w in the two pans right and left respectively, the ratio of the arms is the square root of the ratio of Wj to Wj. The true value of w is the square root of the product w, x Wj. Thus, if when weighed in the right pan, the apparent weight of a body is 37686 grammes, and when weighed in the left, it is 37*S92| T~\/ =1*00125. ^ ^ 3759 2 w=n/37-686 X 37'592=37'635 grammes. The true weight of a body may also be determined in a badly adjusted balance by the following method, known as the method of taring. Place the body in one scale pan and counterpoise it, reading the position of equilibrium of the pointer with as great accuracy as possible ; then, leaving the same counterpoise, replace the body by standard weights, until the position of equilibrium of the pointer is the same as before. The mass which thus replaces the body is evi- dently that of the body, no matter what state the balance may be in. (This is called Borda's method.) (2) To Compare the Weights of the Scale Pans. Let a be the length of the arms supposed equal, s the weight of one pan, and s+<i> that of the other. Weigh a body whose weight is q first in the pan whose weight b s ; let the apparent weight be w. Then interchange the scale pans and weigh q again ; let the weight be V. Then (s+q) tf=(w+s+«*)tf <x(s+«+Q)=(w'+s)tf. Divide each bv a, and subtract; then •i=w'— w— «, or 1 1 8 Practical Physics. (Ch. v. § 13. Thus, weigh the body in one pan ; let its weight be w. Interchange the scale pans and weigh the body again in the other scale pan, but on the same side of the fulcrum ; let the weight be w', then the difference in the weight of the scale pans is i(w'— w). This will be true very approximately, even if the arms be not equal ; for let one be R and the other u Then we have (S + Q)R = (w + S + a>)L (s + a> + Q)R=(w' + s)L A o>=(w' — W— w) — . ^ 'r Now - is nearly unity ; we may put it equal to i +p, where p is very small. 0)=(w' — W — 0)) (i+p) =W' — W — <o + p(w' — W — 0)). But we suppose that o, and therefore w'— w, is very small. Thus p(w'— w— w), being the product of two smaU quantities, may be neglected, and we get <i)=w'— w— 01, or o)=^(w'— w). Experiments, (i) Determine the ratio of the arms of the given balance. (2) Determine the difference between the weights of the icale pans. Enter as below : — (i) Weight in right-hand pan » 37 -686 gms. „ left-hand pan « 37*592 „ r ^ « 1-00125 M w - 37-650 „ (2) Weight in left-hand pan - 37-592 „ M pans interchanged = 37*583 „ .'. Left-hand pan — right-hand pan « -0045 g"^- Cm. V. § 14.] Measurement of Mass. 119 14. Correction of Weighings for the Buoyancy of the Air. The object of weighing a body is to determine its mass, and the physical law upon which the measurement depends Js that the weights of bodies are proportional to their masses, if they are sufficiently near together. Now we have all along assumed that when an adjusted balance-beam was in equihbrium, the force of gravity upon the weights was equal to the force of gravity upon the body weighed, Le. that their weights were equal, and this would have been so if we had only to deal with the force of gra- vity upon these bodies. But the bodies in question were sur- rounded by air, and there was accordingly a force upon each acting vertically upwards, due to the buoyancy of the air ; and it is the resultant force upon the weights which is equal to the resultant force upon the body weighed. But the forces being vertical in each case, their resultant is equal to their difference ; and the force due to the displacement of air by the body is equal to the weight of the air displaced, ie. it bears the same ratio to the weight of the body as the specific gravity of air does to the specific gravity of the body ; while the same holds for the weights. Thus, if w be the weight of the body, <r its specific gravity, and X the specific gravity of air at the pressure and tempera- ture of the balance-case, the volume of air displaced is w/cr and its weight wX/cr (p. 121). Hence the resultant force on the body is w/^i j ; similarly, if <k> be the weights, and p their density, the force on the weights is ^(i — \ These two are equal, thus «»fi — ') / X X\ w=.^ — ^-«(i— +-) approximately, since in general - is very smalL I20 Practical Physics, [Ch. V. § 14. The magnitude of the correction for weighing in air depends therefore upon the specific gravities of the weights, the body weighed, and the density of the air at the time of weighing, denoted by p, <r, and X respectively. The values of p and <r may be taken from the tables of specific gravities (tables, 17, 80) if the materials of which the bodies are com- posed are known. If they are not known, we must determine approximately the specific gravity. We may as a rule neglect the effect of the buoyancy of the air upon the platinum and aluminium weights, and write for /), 8*4, the specific gravity of brass, the larger weights being made of brass. The value of X depends upon the pressure and temperature of the air, and upon the amount of moisture which it con- tains, but as the whole correction is small, we may take the specific gravity of air at 15** C. and 760 mm., when half- saturated with moisture, as a sufficiently accurate value of X. This would give X=*ooi2. Cases may, however, arise in which the variation of the density of the air cannot be neglected. We will give one instance. Suppose that we are determining the weight of a small quantity of mercury, say 3 grammes, in a glass vessel of considerable magnitude, weighing, say, 100 grammes. Suppose that we weigh the empty vessel when the air is at 10** C and 760 mm., and that we weigh it with the mercury in at 15** C. and 720 mm. deducing the weight of the mercury by subtracting the former weight from the latter. We may neglect the effect of the air upon the weight of the mercury itself, but we can easily see that the correction for weighing the glass in air has changed in the interval between the weighings from 22 mgm. to 20*5 mgm. The difference between these, 1*5 mgm., will appear as an error in the calculated weight of the mercury, if we neglect the variation in density of the air, and this error is too considerable a fraction of the weight of the mercury to be thus neg- lected. Ch. v.] Measurement of Mass. 121 Experiment, Determine the weight in vacuo of the given piece of platinum. Enter results thus : — Weight in air at I5°C and 760 mm. with brass weights 37-634 gm. Specific gravity of platinum 21*5. Weight in vacuo, 37 •632. DENSITIES AND SPECIFIC GRAVITIES.* Definition i. — ^The density of a substance at any tem- perature is the mass of a unit of volume of the substance at that temperature ; thus the density of water at 4** C is one gramme per cubic centimetre. Definition 2. — ^The specific gravity of a substance at any temperature is the ratio of its density at that tempera- ture to the density of some standard substance, generally the maximum density of water (i.e. the density of water at 4° C). Definition 3. — ^The specific gravity of a body is the ratio of the mass of the body to the mass of an equal volume of some standard substance, generally water at 4^ C It evidently follows from these definitions that, if p be the density of a substance, o- its specific gravity, and <u the maximum density of water, p^^a-u}, and if m be the mass of a body consisting of the substance, whose volume is v, then MsvpsVo-o)^ and the mass of a volume of water equal to ' It is unfonunate that in many physical text-books the terms * density * and ' specific gravity ' are used synonymously, the former being generally employed for gases and liquids, the latter for solids. It is miite evident that there are two very distinct ideas to be repre- sented, namely (i) the mass of the unit of volume, a quantity whose numerical value depends of course on the units chosen for measuring masses and volumes ; and (2) the ratio of the mass of any volume to the mass of an equal volume of water at 4^ C ; this quantity being a ratio, is altogether independent of units. There being now also two names, ' density ' and * specific eravity', it seems reasonable to assign the one name to the one idea and the other name to the other idea, as suggested by Maxwell, 'Theory of Heat' (ed. 1872, p. 82). When there is no danger of confusion arising from using the term density when specific gravity is meant, there may be no harm in doing io, ^t beginners should be careful to use the two words strictiy In the senses here defined. ^22 Practical Physics. [Ch. v. % 14. the volume of the body = vai. The maximum density of water is i ^amme per cubic centimetre. If we use the gramme as the unit of mass, and the cubic centimetre as the unit of volume, the numerical value of m is unity and the equations we have written become p'=^<r and m=v <r. Thus, the numerical value of the density of a substance on the C.G.S. system of units is the same as the number which expresses the specific gravity of the substance, this latter being of course a ratio, and therefore independent of units. And for the CG.S. system of units, moreover, the numerical value of the mass of a body is equal to the number which expresses its volume multiplied by its specific gravity. These relations are only true for the CG.S. system, and any other systems in which the unit of mass is the mass of the unit of volume of water at 4*^ C. ; but whatever be the system, the density of water at 4" C. is accurately known, although its numerical value may not be unity. Hence, in order to calculate the volume of a body whose mass is known, or vice versd^ we require only to know its specific gravity, and hence the practical importance of determinations of specific gravity. It is generally an easy matter to determine experimentally the ratio of the mass of a body to the mass of an equal volume of water at the same temperature, but it would not be easy or convenient always to keep the water at its temperature of maximum density, throughout the experi- ment The densities of bodies are therefore not usually experimentally compared directly with the maximum density of water in determining specific gravities, and the necessity for doing so is obviated by our knowing with great accu- racy the density of water at different temperatures, (this is given in table 32) ; so that we are enabled, when we know the mass of a volume of water at any temperature, to calculate from the table the mass of the same volume at 4®C., and thus obtain the specific gravity required. We proceed to describe some of the practical methods in creneral use. Ch. V. § 15.] Measurement of Mass. 123 15. The Hydrostatio Balance. llie specific gravity of a substance is determined by the hydrostatic balance by weighing the substance in air, and also in water. One scale pan is removed from the balance, and replaced by a pan suspended by shorter strings from the beam. This pan has a hook underneath, and from the hook the sub- stance to be weighed is suspended by a piece of very fine wire. (i) To determine the Specific Gravity of a Solid heavier than Water, We must first make sure that the beam is horizontal when the balance is loaded only with the wire which is to carry the substance. Turn the key or lever gently to release the beam ; the pointer will probably move sharply across the scale, showing that one pan is heavier than the other. Fix the beam again, and put shot or pieces of tinfoil into the lighter scale until it becomes nearly equal in weight to the other, then let it swing, and observe a resting-point as in § 12. The weights put in should be so adjusted that this resting-point may be near the centre of the scale. Do not counterpoise with weights which you may subse- quently require in order to weigh the object Hang the object whose specific gravity you require — a piece of copper suppose — by the fine wire from the hook above mentioned, and weigh it twice or three times by the method of oscillations (§ 12). Let its weight be 11*378 grammes. Fill a vessel with distilled water, and bring it under the end of the beam so that the copper may dip completely into the water. Be careful that no air-bubbles adhere to the copper ; if there be any, remove them by means of a small brush or feather, or a fibre of glass. It is well to use water that has 124 Practical Physics. [Ch. v. § 15. been freed from dissolved air either by boiling or by means of an air-pump. Any very small bubbles not easily re- movable by mechanical means will then be dissolved by the water. Be careful also that the wire which supports the copper cuts the surface of the water only once ; there is always a certain amount of sticking, due to surface tension, between the wire and the surface of the water, and this is increased if a loose end of the wire be left which rises through the surface. To completely avoid the effect of surface tension the diameter of wire should not be greater than '004 inch. Weigh the copper in the water ; it will probably be found that the pointer will not oscillate, but will come to rest almost immediately. Observe the resting-point, and by turning the key set the beam swinging again, and take another observa- tion. Do this four times, and take the mean. Add some small weight, say 'oi gramme, to the weight, and observe another resting-point, and from these observa- tions calculate, as in § 12, the weight of the copper in water; it will be abput 10*101 grammes. Observe at the same time the temperature of the water with a thermometer. Suppose it is 15°. Then it follows that the weight of the water displaced is ii"378— lo'ioi grammes, or 1*277 gramme. Now the specific gravity of a substance is equal to weight of substance weight of equal vol. water at 4*^0.* In all cases, if we know the weight of a volume of water at /", we can find its weight at 4° C, by dividing the weight at f by the specific gravity of water at f*. Thus, weight at 4" = ^^^^^^ ^.^ ^ . » specific gravity at /• The specific gravity of water at f may be taken from -ble (32). Ch. V. § 15.1 Measurement of Mass. 125 In this case, the weight of the equal volume of water at i5^C b 1*277 grai^nae, and the specific gravity of water at IS** is '99917. /• The weight of the equal volume of water at 4** C. •99917 Thus, the specific gravity of copper It is well to pour the water into the beaker or vessel that is to hold it, before beginning the experiment, and leave it near the balance, so that it may acquire the temperature of the room. If greater accuracy be required, we must free the water used from air. This can be done by putting it under the receiver of an air-pump and exhausting, or by boiling the water for some time and then allowing it to cool. We have neglected the effect of the wire which is im- mersed in the water ; we can, if we need, correct for this. We have also neglected the correction to the observed weight, which arises from the fact that the weights used displace some air, so that the observed weight in air is really the true weight minus the weight of air displaced. (2) To determine tJu Specific Gravity of a Solid lighter than Water. If we wish to find the specific gravity of a solid lighter than water, we must first weigh the light solid in air, then tie it on to a heavier solid, called a sinker, whose weight and specific gravity we know. The combination should be such that the whole will sink in water. Let w and o- be the weight in air, and the specific gravity of the light solid — a piece of wax, for instance — w', fg* corre- sponding quantities for the sinker, w, v for the combina- 126 Practical Physics. [Ch. v. § 15. don ; «/, w the weights in water of the sinker and the combination respectively. I'hen, using C.G.S. units, w/<r represents the volume of the wax, w'/cr' that of the sinker, w/<r that of the combina- tion. Since the volume of the wax is equal to that of the combination minus that of the sinker, we get w_w w' But, with the proper temperature corrections, w _ _ and =~=W — ttf w — — or remembering that wssw+w' w o-= =■ w— a/+tt/* w, a/, w can each be observed, and thus the specific gravity of the wax determined If it is convenient to tie the sinker so that it is immersed while the solid itself is out of the water, the following method is still simpler. Weigh the solid in air and let its weight be w. Attach the sinker below the solid, and weigh the com- bination with the former only immersed. Let the weight be Wj. Raise the vessel containing the water so that the solid if immersed as well as the sinker, and let the weight be w,. Ch. V. § 15.] Measurement of Mass, 127 Then, if the temperature of the water be /**, the specific gravity required X specific gravity of water at f* w,— w (3) To determine the Specific Gravity of a Liquid, Weigh a solid in air ; let its weight be w. Weigh it in water ; let the weight be Wj. Weigh it in the liquid ; let its weight be w^ The liquid must not act chemically on the solid, w— w, is the weight of water displaced by the solid, and w— Wj is the weight of an equal volume of the liquid. Thus, the specific gravity of the liquid at o®, if it expand by heat equally with water, and if the temperature of the two observations be the same, is the ratio of these weights. To find the specific gravity of the liquid at the tempera- ture of the observation, t* say, we must multiply this ratio by the specific gravity of water at the temperature at which the solid was weighed in water ; let this be f. Hence the specific gravity of the liquid at r° = ? X specific gravity of water at /**. W— Wj x- o / Experiments, (1) Determine the specific gravity of copper. (2) Determine the specific gravity of wax. Enter results as below, indicating how often each quantity has been observed. (l) Specific gravjjty of copper. Weight in air . . . il'378 gm. (mean of 3) Weight in water . . . lo-ioi gm. (mean of 3) Weight of water displaced . 1*277 gm. Temperature of water . • 15° C Specific gravity . • . 8*903 128 Practical Physics. (Ch. V. § 15. (2) Specific gravity of wax. Using the piece of copper (f ) as sinker. Weight of wax in air (w) • Weight of sinker (w^) . Weight of combination (w) • Weight of sinker in water {w^ . Weight of combination in water (^ Temperature of water Specific gravity of wax • . 26-653 gm. "•378 ». 38031 ^ lO'IOI „ 9*163 „ 0-965 16. The Specific Gravity BotQe. (i) To determine the Specific Gravity of small Fragments of a Solid by means of the Specific Gravity Bottle. We shall suppose that we require to know (i) the weight of the solid, (2) the weight of the empty bottle, (3) the weight of water which completely fills the bottle, and (4) the weight of the contents when the solid has been put inside and the bottle filled up with water. Strictly speaking, if the weight of the solid fragments can be independently determined, the difference of (4) and (3) is all that is neces- sary, and the weight of the empty bottle is not required ; but in order to include under one heading all the practical details referring to the specific gravity bottle we have added an explanation of the method of obtaining or allowing for the weight of the bottle. The student can easily make for himself the suitable abbreviation if this is not required. We shall also suppose the temperature to be the same throughout the experiment If it consists of only a few fragments of considerable size we may find the weight of the solid by the method ot oscillations; let it be 5*672 grammes. Dry the bottle thoroughly before commencing the experiment The necessity of drying the interior of vessels occurs so firequently in laboratory practice, that it will be well to men- Ch. V. § 16.] Measurement of Mass, 129 tion here the different methods which are suitable under different circumstances in order that we may be able to refer to them afterwards. We may take for granted that all the water that can be removed by shaking or by soaking up with slips of filter paper, has been so got rid ofl An ordinary bottle or flask can for most purposes be suf- ficiently dried by drawing air through it by means of a tube passing to the bottom of the bottle and connected with an aspirator or the aspirating pump referred to in the note (p. 89), and at the same time gently warming the bottle by means of a spirit lamp. If there be any considerable quantity of water to be got rid o^ the process can be considerably shortened by first rinsing out the bottle with alcohol. If more careful drying is necessary, as, for mstance, for hygrometric ex- periments, the mouth of the vessel should be closed by a cork i)erforated for two tubes, the one opening at one end and the other at the other end of the vessel, and a current of perfectly dry air kept passing through the vessel for some hours. The air may be dried by causing it to pass first through U-tubes filled with fused chloride of calcium, which win remove the greater part of the moisture, and finally through a tube containing phosphoric anhydride or frag- ments of ignited pumice moistened with the strongest sulphuric acid. If there be no opening in the vessel sufficiently large to allow of two tubes passing, the following plan may be adopted : — Connect the tube which forms the prolongation of the plug of a three-way tap ' with an air-pump. The water air-pump before referred to is very convenient for the purpose if there be a sufficient head of water on the water- * A three-way tap Is a simple, but in many ways very useful, con- trivance. In addition to the two openings of an ordinary tap, it has a third, formed by a tabular elongation of the plug, and communicating whh that part of the conical face of the plug which is on the same cross- lection as the usual holes, but at o^e end of a diameter perpendicular to the line joining theuL Such taps may now be obt^ed from many of the glass-blowers* K 130 Practical Physics, [Cn. V. § id. supply to give efficient exhaustion. Connect the other openings of the tap with the vessel to be dried and the dry- ing tubes respectively. Then, by turning the tap, connection can be made alternately between the pump and the vessel and between the vessel and the drying tubes, so that the vessel can be alternately exhausted and filled with dried air. This process must be repeated very many times if the vessel is to be completely dried. Having by one of these methods thoroughly dried the bottle, place it on one of the scale pans of the balance, and counterpoise on the other either with the brass weight provided for the purpose, or by means of shot or pieces of lead. Observe the resting-pbint of the pointer by the method of oscillations, taking two or three observations. Meanwhile a beaker of distilled water, which has been freed from sur either by boiling or by being enclosed in the exhausted receiver of an air-pump, should have been placed near the balance, with a thermometer in it, in order that the water used may have had time to acquire the temperature of the room and that the temperature may be observed. Fill the bottle with the water, taking care that no air- bubbles are left in. To do this the bottle is filled up to the brim, and the stopper well wetted with water. The end of the stopper is then brought into contact with the surface of the water, taking care that no air is enclosed between, and the stopper pushed home. All traces of moisture must be carefully removed from the outside of the bottle by wiping it with a dry cloth. Observe the temperature of the water before inserting the stopper ; let it be 15^ C. The bottle should be handled as little as possible, to avoid altering its temperature. Replace the bottle on the scale pan, and weigh ; let the weight observed be 24*975 grammes. This weighing, like every other, should be done twice or ihree times, and the mean taken. This is the weight of the water in the bottle only, for we Ch. V. § 16] Measurement of Mass. 131 have supposed that the bottle has been previously counter- poised Open the bottle and introduce the small fragments of the solid which have been weighed, taking care to put all in. Again fill the bottle, making sure by careful shaking that no air-bubbles are held down by the pieces of the solid ; if any are observed, they must be removed by shaking or by stirring with a clean glass rod ; or, if great accuracy is re- quired, by placing the bottle under the receiver of an air- pump and then exhausting, Replace the stopper, carefully wiping off all moisture, and weigh again, twice or three times ; let the weight be 27*764 grammes. This is clearly the weight of the substance -f the weight of the bottleful of water - the weight of water displaced by the substance. Thus the weight of water displaced is equal to the weight of the substance -I- the weight of the bottleful of water — 27*764 grammes = 30-647— 2 7 '764= 2 "883 grammes. Now we require the weight of water which would be displaced were the temperature 4**C. ; for the specific gravity of a substance is equal to weight of substance weight of equal vol. water at 4* but the weight of any volume of water at 4** _ weight of equal vol. at f* "specific gravity water at f* Thus the specific gravity of the substance weight of substance ^ ^^^ g^^ ^^^^^ at t\ weight of equal voL water at /** 132 Practical Physics, [Ch. v. % 16. Taking from the table (32) the specific gravity of water at 15'', we find the specific gravity of the substance to be 5-1? X -9991 7= I -966. If greater accuracy be required, we must free the water used from air by boiling or the use of the air-pump. We should also require to correct the weighings for the air displaced. (2) To find the Specific Gravity of a Powder, The process of finding the specific gravity of a powder is nearly identical with the foregoing. The only modifica- tion necessary is to weigh the powder in the bottle. The order of operations would then be — (1) Counterpoise the dry bottle. (2) Introduce a convenient amount of the powder, say enough to fill one-third of the bottle, and weigh. (3) Fill up with water, taking care that none of the powder is floated away, and that there are no air-bubbles, and weigh again. If it be impossible to make all the powder sink, that which floats should be collected on a watch-glass, dried, and weighed, and its weight allowed for (4) Empty the bottle, and then fill up with water and weigh again The method of calculation is the same as before. (3) To determine the Specific Gravity of a Liquid by the Specific Gravity Bottie. Fill the bottle with water, as described above, and weigh the water contained, then fill with the Hquid required, and weigh again. Each weight should of course be taken twice. The ratio of the two weights is the specific gravity of the hquid at 4** C. if it expand by heat equally with water. If we require the specific gravity of the liquid at the temperature of the experiment, we must note the tempera ture of the water, and reduce its weight to the weight of an Ch. V. S 16.1 Measuntmeni of Mass. 133 equal volume at 4^ C. ; that is, we must multiply the above ratio by the specific gravity of water at the temperature of the observation. Thus, the specific gravity of a liquid jreight of liquid ^ ^^ ^^ ^^j^, ^j ^ weight of equal voL water at f^ Experiments. (i) Determine the specific gravity of the given solid. (2) Determine the specific gravity of the given liquid. Enter as below, indicating the observations made of each quantity : — fi) Specific gravity of solid Weight of solid .... 5*672 gm. Weight of water in bottle . . 24^975 gm. Weight of water with solid . 27*764 gm. Temperature, 15** C. Specific gravity, 1*966. (2) Specific gravity of liquid. Weight of water in bottle . . 24-975 gm. Weight of liquid 23-586 gm. Temperature . , . i5<>C. Specific gravity of liquid . . *943a 17. Hioholson's Hydrometer. This instrument is used (i) to determine the specific gravity of small solids which can be immersed in water ; (2) to determine the specific gravity of a liquid. (i) To fitid the Specific Gravity of a Solid. Taking care that no air-bubbles adhere to it, place the hydrometer in a taU vessel of distilled water recently boiled, and put weights on the upper cup until it just sinks to the oiark on the stem. To avoid the inconvenience caused by the weights falling into the water, a circular plate of glass is provided as a cover 1 34 Practical Physics. [Cii. v. { 17. for the vessel in which the hydrometer floats. This has been cut into two across a diameter, and a hole drilled through the centre, through which the stem of the instrument rises. It will generally be found that with given weights on the cup the hydrometer will rest m any position between certain limits; that there is no one definite position of flotation, but many. The limits will be closer together and the experiment more accurate if the surface of the instru- ment, especially that of the stem, be thoroughly clean and free from grease. It is well therefore carefully to rub the stem and upper part of the bulb with some cotton-wool soaked in methylated spirit Suppose now it is floating with the mark on the stem just below the surface. Take off* some weights until the mark just rises past the surface ; let the weights then on be 8*34 grammes. Put on weights until the mark just sinks below the siulace, and then let the weight be 8*35 grammes. Do this several times, and take the mean as the weight re- quired to sink the mark to the surface. Let the mean be 8*345 grammes. Remove the weights and put the solid in the upper cup. Then add weights until the mark again just comes to the surfoce, estimating the weight required as before. Let this be 2*539 grammes. The weight of the solid in air is the diflerence between these, or 5*806 grammes. Now put the solid in the lower cup* and weights in the upper one until the mark sinks to the surface. Estimate as before. Let the mean of the weights be 5*462 grammes. The difference between this and the weight 8*345, put on originally to sink the hydrometer, gives the weight in water. Thus, the weight in water = 2*883 grammes. And the weight of water displaced = weight in air — weight in water = 2*923 grammes. • If the solid be lighter than water it must be listened down to the cnp either by a wire or by being enclosed in a cage fixed to the instm- mcDt. Ch. V. § 17.] Measurement of Mass. 135 The specific gravity, therefore, referred to water at the temperature of experiment 2923 To determine the true specific gravity — water at 4® C. being taken as the standard — we must multiply this number by the specific gravity of the water at the time of the ex- periment. This may be taken from the table (32), if we know the temperature. Thus, we must observe the temperature of the water at the time of the experiment. Let it be 15^ Then the specific gravity required =1*987 X '99917=1*985 approximately. (2) To determine the Specific Gravity of a Liquid. Let the weight of the instrument itself be 11*265 grammes. This must be determined by weighing it in a balance. Place it in the water, and put weights on the upper pan onnl it just floats up to the mark on the stem. Let the weight be 8*345 grammes. This of course must be estimated as in experiment (i). ^ The sum of these two weights is the weight of a volume of water equal to that of the instrument up to the mark on the stem. Thus, the weight of this volume of water is 19*610 grammes. Now place the instrument in the liquid and add weights till the mark is just in the surface. Let the weight be 9*875 grammes. Then the weight of the volume of liquid displaced is 11*265-1-9*875 or 21*140 grammes. The specific gravity of the liquid referred to water at the temperature of the experiment is therefore ii:i4£=ro78. i9*6io 136 Practical Physics, [Cii. V. 5 17. Let the temperature of the water be i5**C ; that of the liquid ii'5®G Then the specilic gravity of liquid at 115° C is 1-078 X '99917 =1-077. Experiments. (i) Determine the specific gravity of sulphur by Nicholson's Hydrometer. (2) Make a 20 per cent, solution' of common salt in water, and determine its specific gravity by Nicholson's Hydrometer Enter results thus : — {a) Specific gravity of sulphur. Mean weight required to sink the hydrometer to the mark 8-345 gms. Mean weight required to sink the hydrometer with sulphur on upper pan .... 2*539 ^ Mean weight required to sink the hydrometer with sulphur on lower pan . . . 5 '462 „ Temperature of the water, 15" C. Sp. gr. of sulphur « 1*985. (b) Specific gravity of salt solution. Weight of salt used 539*o gu^s. Weight of water used 21560 „ Weight of hydrometer 11 '265 ^ Weight required to sink the instrument to the mark in water at 15^ 8*345 „ Weight required to sink instrument in solution atii***5C 9-875 II Specific gravity of solution • • . • 1*077 w 18. Jolly's Balance. The apparatus consists of a long spiral spring carrying a pan into which weights or the object to be weighed can be put ' A 20 per cent solution b one which contains 20 parts by weight of salt in loo parts of the solution. It may therefore be made by adding the salt to water in the proportion of 20 grammes of salt to 80 grammes of water. Ch, V. § i8.] Measurement of Mass, 137 From this there hangs, by a fine thread, a second pan which is always kept immersed in water. Behind the spring is a millimetre scale engraved on a strip of looking-glass, and just above the pan is a white bead, which can be seen directly reflected in the glass. By placing the eye so that the top of the bead just appears to coincide with its own image, the division of the scale which is opposite to the top of the bead can be read with great accuracy. (i) To weigh a small Body and find its Specific Gravity. Place the object to be weighed in the upper pan, taking care that the lower pan is well below the surface of the water, and that the vessel in which the water is, is suffi- ciently large to allow the pan to hang clear of the sides. Note the division of the scale which coincides with the top of the bead. Suppose it is 329. Remove the object from the pan and replace it by weights until the bead occupies the same position as before. Let the weights be 7*963 grammes. It may be impossible with given weights to cause the bead to come to exactly the same position. Thus, we may find that 7*963 gms. causes it to stand at 330, while 7*964 gms. brings it to 327*5. The true weight lies between these two ; and the addition of '00 1 gramme lowers the bead through 2*5 mm. We require the bead to be lowered from 330 to 329— that is, through i mm. We must therefore add to our weight — of *ooi gramme, or 0*0004 gramme. The true weight then would be 7*9634 grammes. The water should be adjusted so that its surface is above the pomt of junction of the three wires which carry the lower pan. Next place the small object in the lower pan, and put weights into the upper till the bead again comes to the 138 Practical Physics. [Ch. V. § 18, same point on the scale. Let the weights be 3*9782 grammes. This is clearly the weight of the water displaced by the object, and its specific gravity referred to water at the tem- perature of the observation is therefore ?_2_34 Qj. 2 -002. 3-9782 To obtain the true specific gravity, we must multiply this by the specific gravity of the water at the temperature of the observation. Let this be 15^. The specific gravity of water at 15^ is '999179 so that the specific gravity of the solid is 2*002 X *999i7> or 2*000. (2) To determine tJie Specific Gravity of a Liquid. Take a small solid which will not be acted on by the liquid, and place it in the upper pan. Note the point to which the bead is depressed, die lower pan being in water. Now place the solid in the lower pan and put weights into the upper until the bead comes opposite the same mark« Let the weight be 3*596 grammes. This is the weight of the water displaced by the solid. Remove the water and replace it by the liquid. Put the solid into the upper pan, and note the division opposite to which the bead stands. Let it be 263. Put the solid into the lower pan, and put weights into the upper until the bead comes opposite to 263. Let the weight be 4*732 grammes. This is the weight of the liquid displaced by the solid. Thus, the specific gravity of the liquid =473^=1.316. ^590 '^18 must be corrected for temperature as usual. Cb. V. } 18.1 Measurement of Mass. 139 Experiments. (1) Detennine by means of Jolly's Balance the specific gravity of the given small crystal. (2) Determine by means of Jolly's Balance the specific gravity of the given liquid. Enter the results thus : — (i) Specific gravity of crystal Scale reading with the crystal in the upper pan 329 mm. Weight required to bring the bead to same position 7*9634 gms. Weight required with crystal in lower pan . 3*9782 », Temperature of water 15^ C. Sp. gr. of crystal 2*000. (2) Specific gravity of liquid. Scale reading with solid in upper pan, lower pan in water . .... 329 mm. Weight required to bring the bead to the same reading with the solid in water. . . 3*596 gms. Scale reading with the solid in the upper pan, lower pan in the liquid .... 263 mm. Weight required to bring the bead to the same reading with the solid in the liquid . . 4*732 gms. Temperature of the water 15^ C. Specific gravity of liquid "1*315. 19. The Common Hydrometer. The specific gravity of a liquid may be most easily determined to within 0*1 percent by the use of the common hydrometer. This instrument consists of a glass bulb with a cylin- drical stem, loaded so that it floats in any liquid whose specific gravity lies within certain limits, with the stem vertical and partly immersed. The depth to which it requires to be immersed in order to float is defined by the condition that the weight of the liquid displaced is equal to the weight of the hydrometer. For any liquid, therefore, I40 Practical Physics. [Ch, V. § 19. within the limits, there b a definite point on the stem to v^hich the instrument will sink, depending on the specific gravity ; and the stem can be graduated in such a manner that the graduation reading gives the specific gravity at once. This is generally done by a scale attached to the inside of the stem, and hence all that has to be done to determine the specific gravity of a liquid is to fioat in it a suitable hydrometer, and take the scale reading at the sur- face. The temperature correction is to be allowed for as usual An instrument sensitive to such slight variations of density as 0*1 per cent, would require to have too long a stem if used for the whole range of density commonly occurring. Hydrometers are, therefore, usually obtained in sets of three or four, each suitable for one portion only of the range. The case in which they are kept contains a long cylindrical vessel, which is convenient for floating them in and also a thermometer. The hydrometers, vessel, and thermometer should be carefiiUy washed and dried before replacing them in the case. The graduation of the scale is a comparatively difficult matter, as equal increments in the length of the stem immersed do not correspond to equal difierences of density. The scales are graduated by the instrument-makers, and we require to be able to test the accuracy of the graduation. We can do this by taking the hydrometer readings in liquids whose specific gravities are known. Distilled water would naturally be a suitable one for the purpose. The hydrometer when floating in distilled water at i5^C should read 0*999. The specific gravity of any other suitable liquid could be determined by one of the methods already de- scribed. The following experiment, however, serves as a very instructive method of comparing the density of any liquid with that of water, and it is, therefore, suggested as a «"eans of testing the accuracy of the hydrometer scale. - a ^ Ch. V. j 19.1 Measuremmt of Mass. 141 To ampare tht Densitks of two Liquids by the Aid of tht Katlietomtter. \l we have a. U tube (fig. 1 1) and fill one leg with one liquid standit^ up to the level p, and the other with a second up to the level q, and if r be the ^"=- "■ common surface of the liquids in the two legs p B, Q R, their densities are inversely proportional to the vertical distances be- tween p and R, Q and r,' These can be acctuatelj measured by the kathetoraeter, and the densities thus compared. If the kathetometer be not available, the heights may be measured by scales placed beliind the tubes, which are read by a telescope placed at a distance and roughly levelled for each observation. This arrangement supposes that the two liquids do not mix. The following apparatus is therefore more generally available : — A B c, D B F are two U tubes, the legs b c, d e being the shorter. These legs are connected together by a piece of india-fubber tubing c c d. One liquid is poured into the tDbe ^'°" "' A B, and then the other into the tube p e. This, as it runs down the tube, compresses the air below it, thus in- creasing the pressure on the surface of the first liquid, and forcing it up the leg BA. The quantity poured into f e must not be sufficient to rise over the end D of the tube. Now pour more of the first liquid into A B. This forces up the levd of the liquid in s p, and after one or two repetitions of this ■ See below, chip. *u. p. 197. o 143 Practical Physics. jch. V. % 19. operation the levels of the liquid in one tube will be at a and c, those in the other being at f and d. The pressure at c and d, being that of the enclosed air, is the same. The excess of the pressure at c above the atmospheric pressure is due to a column of liquid of height equal to the vertical distance between a and c, that at nis due to a column of the second liquid of height equal to the distance between p and D. These distances can be observed by the kathetometer, and the densities of th^ two liquids are inversely propor- tional to them. The surface of the liquids in the tubes will be curved, owing to capillary action. In measuring, either the bottom or the top of the meniscus, whichever be most convenient, may be observed, but it is necessary to take the same at each end of the colunm. The bottom will, if the liquid wet the tube, give the more accurate result. It is well to hang up behind the tubes a sheet of white or grey paper, to afford a good background against which to see the liquids. It is important that the temperature should remain the same during the experiment; for if it increase the pressure in the portion cod increases, and the air there expands, thus forcing up the columns of liquid. We may avoid the difficulty this causes by the following method of taking the measurements : Observe the height of a, then the height of c, and finaUy the height of a again. llien, if the temperature has changed uniformly and the intervals between the successive measurements have been the same, the mean of the two observed heights of a will give its height at the time when the observation of the height of c was made, and the difference between these two, the mean of the observed heights of a and the height of c, will give the true height of the column. Ch. V. § 19.] Measurement of Mass, 143 If one liquid be water at a temperature, say, of 15^ C, the ratio of the two heights gives us the specific gravity of the second liquid, for its temperature at :he time of the observa- tion, referred to water at 15® C If we wish to find the true specific gravity of the liquid at the temperature of the observation, 15** C, we must multiply the above ratio by the specific gravity of water at i5^C Suppose the second liquid is also at 15^ C, and that its coefficient of expansion by heat does not differ greatly from that of water. Then the same ratio gives us the specific gravity of the liquid at 4** C referred to water at 4** C, or the true specific gravity of the liquid at 4^ C. without any correctioa Experiment. — Determine the specific gravity of the given liquid by means of the hydrometer, testing the accuracy of the results. Enter results thus : — Specific gravity by hydrometer I'sSj. Tube AC water ; tube dp liquid. Hdgfat of A Mean Hdgl.t ol C ^^^' 1 23-522 86-460 23-535 ) Difference 62*938 Temperature of the water, i ^Q. Height of F Mean Height of D alSs) ^5-6.8 84-365 Difference 48-747 Temperature of the liquid 13-5 C Specific gravity of liquid - ^^22- x •999x7 -» 1*290 144 Practical PhysHs. [Cb. v.» 5 it CHAPTER V.' UEASUKEHENT OF VELOCITY AND ACCELERATIOK. B. To Heamre the Telocity of a Peadolom. When a body is moving uniformly in a straight line its velocity is measured by measuiing the distance it traverses in a measured interval. When the velocity is changing, this method is no longer applicable. The measurement of the distance traversed in a known interval gives only the mean velocity for that interval, but by maldng the interval sufficiently short the result represents adequately the actual Pi_. ^ velocity during the inter- val. We shall in this chapter shew bow by making use of the very short intervals corre- sponding to the time of vibration of a tuning-fork a &.ir measure of the ve- locity of a moving body can be obtained, and shall further shew how by geometrical methods upon a true scale-diagram ' of the path of a moving body the velocity and acceleration of the body can be determined. The velocity of the pendulum — is not uniform at any point of its path ; but when near its lowest position it has a maximum value which varies very slowly. This maximum value may be found thus in the case of a heavy pendulum Cb, V.» § B.] Measurement of Velocity ^ &c. 145 (a b, fig. vi) mounted so as to move in one plane. A glass plate c is attached to it, and a piece of smoked card or metallic paper is fixed on the plate in such a way that the plane of the card is the plane in which the pendulum moves. D is an adjustable rod fitted with a small movable hook, to which the pendulum can be secured in any desired posi- tion. On pulling the hook down by a string attached to it, the pendulum is released and starts swinging. The card in its motion is just touched by a light metallic pointer ; if the pointer be fixed, an arc of a circle is traced on the card by the pointer as the plate swings past This pointer is attached to the prong of a large tuning-fork, which vibrates in a vertical plane. The tuning- fork is not shewn in the figure, but rests on the stand below the pendulum. If when the tuning-fork is set in motion the pendulum is again started, the pointer traces out on the card a sinuous curve cutting the circle in a number of points— i, 2, 3, 4, &c. Each point corresponds to the passage of the fork through its position of equili- brium. Now the characteristic property of the tuning-fork is that the interval between successive passages through the equilibrium position is constant. This constant value is not greatly altered by the friction between the pointer and the card. Let us suppose its value is known for the fork in question, and that it is ijn of a second ; n may con- veniently be about 60, so that the fork is making 60 vibra- tions per second. Then the distances between the successive points i, 2, 3, &a, are the distances moved over by the pendulum in successive »ths of a second. The distances will vary slightly, but towards the centre of the trace, where the pendulum is moving at its maximum rate, the variations will be small and the waves longer than at either end. Select some of these waves of maximum length, and measure a number of them with a fine scale and dividers, or by the aid of reading-microscopes. Let the L 1 46 Practical Physics, [Oa. V.* § b. length be x centimetres ; this distance was traversed in I /ff of a second, and thus the mean velocity for that period \& nx centimetres per i second. Measure the vertical distance h between the position of a point p on the pen- dulum at the bottom of its swing and the point from which p started, and by a series of observations verify the law that i? is proportional to h. When the velocity of the pendulum has been deter- mined thus by the aid of a known fork, the same ap- paratus may be used to determine the period of a tuning- fork, for if we repeat the experiment using a fork of un- known frequency «', and if x! be the lengths of the waves as measured, then x^ti = velocity of pendulum at the lowest point of its swing = xn, and hence «' = nx/x^. Again, the trace of the fork may be used so as to measure in a similar way the velocity at other parts of the swing, and thus the rate of change in velocity can be deter- mined. But the rate of change of velocity is the accelera- tion, and we can thus verify die fact that the acceleration is proportional to the displacement from the equilibrium position. To obtain the acceleration plot a curve on squared paper, taking the times as abscissae and the velocities as ordinates (see p. 50). The velocity is proportional to the length of the waves. We may thus take the horizontal divisions of the paper to represent the period of the fork — one-sixtieth of a second suppose — and at the end of each division draw a vertical ordinate proportional to the corre- sponding measured wave-length. We thus obtain a curve such as APQ (fig. vii), amn being the time line. Let p M, Q N be ordinates at two instants close together. Draw p R parallel to a m n to meet q n in r ; then r q is the increase in velocity in time mn, and the ratio qr/mn measures the average acceleration or rate of increase of velocity during that interval. Now when q is very near to x> rij> becomes ultimately p t, the tangent to the qurve at ratio Q r/m n measures tan p t a or tan 0. Thu5« Cr. v.* § B.] Measurement of Velocity^ &e. M7 the acceleration at each point is measured by the tangent of the angle which the tangent to the velocity curve makes with the time axis Fig. vS. A curve may therefore be plotted in a similar way for the acceleration, and will be found similar to the velocity curve, but with the maxima in a different position. Now in the velocity curve the space described in any interval is represented by the area between the curve, the time line, and the two ordinates at the ends of the interval Cal- culate the value of this area for the times i, 2, 3 . . ^ reckon- FiG. viU. ing from some convenient instant, and determine the values of the acceleration, or tan 0, at the same instants. Then plot a third curve with the values of the acceleration as ordinates and the distances as abscissae. It will be found L 2 148 Practical Physics. [Ch. v.* § c to be a straight line, as in fig. viii. Thus p m, the accelera- tion after moving a distance om, is proportional to mb, the distance from some point a This point b represents the equilibrium position. C. To trace the Curve described by a Falling Body and the character of its downward Acceleration. Take a bottle provided with an aperture in one side near the bottom. A small glass nozzle is fitted into this, so that water issuing from the bottle emerges in a horizontal direction. A pipe passes through a cork in the top of the bottle and reaches about two-thirds of the way down. The bottle is partly filled with water. As the water runs out through the nozzle air enters in bubbles through the pipe ; the pressure at the bottom of the pipe is equal to the atmospheric pressure, and hence the pressure of the issuing jet remains constant so long as the surface of the water is above the bottom of the pipe ; thus the stream is steady. Under these circumstances the curve described by the water remains unchanged, and this curve is the same as that described by a falling body projected from the nozzle with the velocity of the issuing jet. Place near the water-jet, and parallel to its plane, a sheet of glass, and at the farthest convenient distance on the other side place a powerful lamp which will throw a shadow of the jet on the glass. Hold up a sheet of paper against the glass, standing on the side of the glass remote from the lamp ; the shadow will be clearly seen, and can be traced on the paper with a pencil. Fix a ruler in a hori- zontal position so that it casts a shadow on the paper, and thus draw a horizontal line on the paper. ^ * A better curve for the purposes of measurement can be obtained by first drawing a parabola on paper and then adjusting the pressure of the water untU the shadow exactly corresponds with the drawn curve. Ch. v.* § cj Measurement of Velocity^ Stc 149 Let A (fig. ix) be the highest point on the curve corre- sponding to the shadow of the nozzle from which the jet issues horizontally Through a draw horizontal and vertical lines A M X, A N y. Let p be any point on the jet, and p m, PN vertical and horizontal. Now the horizontal velocity remains unchanged ; thus, as p moves along the jet, m moves uniformly along the line a x, and we may take a m to represent the time of travelling from a to p. p M is, of course, the vertical distance traversed in this Fiaix. M M' K N time. Now let p' be a neighbouring point, and let p' m' be drawn vertical to meet a x in bi'. Draw p r horizontally to meet p' m' in R. Then p' r is the vertical distance traversed in time m m'. The average vertical velocity during this time is therefore the ratio p' r/m m'. Let p t, the tangent at p, meet a x in t ; then when m m' is very small p' p coincides with p T, and the limiting . value of the ratio p' r/m m' is p m/m t, or the tangent of the angle p t m. This, then, measures the vertical velocity at p. On PM take a point q, such that qm is propor- tional to tan p T M. Then Q m will represent the vertical velocity at p, and corresponding to each point such as p 1 50 Practical Physics. fCn. V.» $ c. a point Q can be found. A curve can be drawn through these points, and this curve will be the vertical velocity- curve for the ^ing body. The simplest method of deter- mming the position of Q is to set off any convenient constant length t k, say a centimetres, from t along t x, and then through k draw K l vertical to meet t p in l ; from L draw LQ horizontal to meet m p produced. Then QM = LK = TKtanPTM = « tan ptbc. Thus QM represents the vertical velocity. Now if the figure be carefully drawn, it wiU be found that the curve traced out by Q is a straight line passing through a. Thus the vertical velocity increases imiformly with the time, and the vertical acceleration is therefore a constant, and is repre- sented by the tangent of q a x. We can represent the results symbolically thus. Let u represent the constant horizontal velocity, v the vertical velocity, / the time from a to p. Let AM=^, PM=>', QM=«, PTM=0, and qax=U^ Then dt dx a Let g be the vertical acceleration. dv u dz «* dz u^ . I ^ dt a dt a dx a ^ If we know the value of g^ this result gives the initial horizontal velocity. It follows from the above results that the curve traced out by p is a parabola. This can readily be verified from the figure, for on measuring the values of p M and p n foi different positions of p, it will be found that p N* is always proportional to p M, and this is the fundamental property r.( » -"-"^bola with its vertex at a and a v for its axis. Ch. V,* S C.1 Measurement of Velocity , &c 1 5 1 The curve may be shewn to possess the other charac- teristic properties of a parabola, and, conversely, some of the known properties of the parabola may be employed to find the focus, axis, and direction of the curve. Thus, if a series of chords be drawn parallel to the tangent at any point p, the diameter, which bisects all these chords, will be a straight line parallel to the axis. If q v be one-half of one of the chords, the property q v* = 4 s p, p v may be employed to determine s p, the distance of the focus from Py in terms of lengths that can be measured in the figure. Determining in this way the value of s p for two points, p„ P3, the focus can be obtained as the intersection of two circles with radii Pj s, Pj s respectively. The axis is the line through the focus drawn parallel to any diameter. The directrix is the locus of intersections of tangents at right angles, the tangent at the vertex is the locus of the foot of the perpendicular from the focus on the tangents, and thus each of these lines can be drawn when the curve ^ only is figured on the paper. ' The curve can be shewn experimentally to be described by a pendulum-bob with a long Y-suspension, when the distance of the bob mm the junction of the strings is one-quarter of the whole vertical distance of the bob from the points of support of the strings ; and also to be the boundary of the shadow of a circle thrown upon a horizontal plane by a poiat of light on a level with the top of the rim of the circle. 1 52 Practical Physics. [Ch. VL § 2a CHAPTER VI. MECHANICS' OF SOLIDS. 20. The Pendnlnm. (i) To determine the Value of % by Observations with the Pendulum. If / be the time of a complete oscillation of a simple pen- dulum whose length is /, and g the acceleration due to gravity, then it can be shewn that (See Maxwell, * Matter and Motion,' Chap. VII.) Thus, We can therefore find the value of g by observing /, the time of a complete oscillation, and /the length of the pendulum. A heavy sphere of metal suspended by a fine wire is, for our purposes, a sufficiently close representation of a simple pendulum. Corrections for the mass of the suspending wire, &c, can be introduced if greater accuracy be required. To observe /, focus a telescope so that the wire of the pendulum coincides • when at rest with the vertical cross- wire. A sheet of white paper placed behind the wire forms a suitable background. Set the pendulum swinging, and note by means of a chronometer or clock the times of some six consecutive transits, in the same direction, of the pen- dulum across the wire of the telescope. To obtain these, the best plan is to listen for the ticks of the clock, and count in time with them, keeping one eye at the telescope. Then note on paper the number of the tick at which each successive transit takes place. Thus, suppose the clock beats half-seconds, we should obtain a series of numbers as follows : — Ch. VI. § aal Mechanics of Solids. 153 No. of transit (i) (a) (3) (4) (s) (6) Time noted, 11 hrs. lomin. 2, 9, 17, 26, 34, 43 ticks. Thus, successive transits in the same direction occur at the following times : — No. of transit (i) («) (3) (4) (s) (6) Time, 11 hrs. 10 min. • . i, 4*5, 8*5, 13, 17, 21*5 sec Wait now for one or two minutes,* and observe again : — Transit (7) (8) (9) (10) (11) (12) Time, 11 hrs. 14 min. . . 9, 13*5, 17, 22, 26, 30 sec. Subtracting the time (i) from (7), (2) from (8), &c., we get the times of a certain large but unknown number of oscillations — vis., 4 min. 8 sec., 4 min. 9 sec., 4 min. 8*5 sec., 4 miiL 9 sec., 4 min. 9 sec., 4 min. 8*5 sec ; the mean of these is 4 min. 8*66 sec So that in 248*66 sec. there is a large whole nimiber of complete oscillations. We have now to find what that number is. From our first series of observations we may see that five complete osciUations occupy 20*5 sec Thus, the time of an oscillation deduced from this series is \ of 20*5 or 4*1 sec ; from the second series \r of 21, or 4*2 sec Thus, die time of a complete oscillation deduced from these two sets of observations is 4*15 sec If this were the true time of an oscillation, it would divide 248*66 sec. exactly. On doing the division, the quotient obtained is 59*92 sec This is very nearly 60, and since there has been a whole number of oscillations in the 248*66 sec the whole number may have been 60, and, in consequence, the time of an oscillation 248*66/60 — i.e. 4*144 sec This method of measuring accurately the time of an oscillation turns upon measuring roughly the time of oscil- lation and then determining the exact number of oscillations in a considerable interval by dividing the interval by the ap- proximate measure of the time of oscillation, and selecting the nearest integer. One important point requires notice. * The rule for detennining the proper interval which ihoald be allowed is given later, p. 154* 1 54 Practical Physics. [Ch. VI. § aa The rough value of the time of oscillation was determined by observing the time of five oscillations with a dock shewing half-seconds. We must therefore consider the observation of the first and sixth transit as each liable to an error of half a second ; that is, the time of the five oscillations is liable to an error of one second, and the calculated time of one is only to be regarded as accurate within 0*2 sea All we can be sure of, therefore, is that the time of an oscillation lies between 3*95 sec and 4*35 sec Now the nearest integer to 248'66/3*95 is 63, and the nearest integer to 248*66 /4"35 is 57 ; hence, without more observations than have been indicated above, we are not justified in taking 60 as the proper integral number of oscillations during the interval All we really know is that the number is one ol those between 57 and 63. In order that there may be no doubt about the proper integer to select, the possible error in the rough value of the time of oscillation, when multiplied by the integer found, must give a result less than half the time of an oscillation; thus in the instance quoted the inference drawn is a safe one, provided 4*15 sec represents the period of one oscilla- tion to the thirtieth of a second. If this be the case the method given above will indicate the proper integer to select as representing the number of oscillations in 248 sec, and thereifore give the time of an oscillation correct to about the 250th of a second. There are two ways of securing the necessary accuracy in the observed time of an oscillation : (i) by making a series of thirty-one transit observations instead of 6, as indicated above ; and (2) by repeating the process sketched, using intervals sufficientiy small for us to be certain that we can select the right integer. Thus, suppose six transit observations are made, the second series must be made after an interval not greater than 20 sec, a third after an interval of 60 sec from the first, a fourth after an interval of 140 sec From the original Ch. VI. § aa] Mechanics of Solids. 155 series a result will be obtained accurate to 0*2 sec. ; with the first and second the accuracy can be carried to o*i sec, with the first and third to 0*05 sec. ; and so proceeding in this way, we can with complete security carry the accuracy to any extent desired. To determine /, we measure the length of the suspend- ing wire by means of a tape, and add one half of the diameter of the bob as measured by the calipers. If the value of gravity is to be expressed in C.G.S. units (cm. per sec. per sec), the length must be given in centi- metres. Thus the values of/ and / have been found. Substituting these in the formula for g^ its numerical value may be found. The value of x may be taken as 3'i42. (2) To compare the Times of Oscillation oj two Pendulums, Metliodof Coincidences, The method is only applicable in the case of two pen- dulums whose periods of oscillation are very nearly in some simple ratio which can be roughly identified. The two pendulums are arranged one behind the other, and a screen is placed in front with a narrow vertical slit. A telescope is arranged so as to view through the slit the nearer of the two wires. The second one is not visible, being covered by the first. Let us suppose that the shorter pendulum vibrates rather more than twice as fast as the longer. Start the two pendulums swinging ; the two wires will appear to cross the slit at different moments. After a few swings they will cross in the same direction at the same moment We may notice that the shorter pendulum, besides executing two oscillations while the longer executes one, gradually gains on the latter, but after a time the two again cross simultaneously in the same direction. Let us suppose that this happens after 12 oscillations of the long pendulum ; then there have been clearly 25 oscillations of the shorter 1 56 Practical Physics. [Ch. VI. 5 aa in the same interval. Thus, the time of oscillation of the shorter pendulum is — - X 4*144, or 1*9891 sea If the longer pendulum had been gaining on the shorter, the latter would have lost one oscillation during the interval, and the ratio of the times of oscillation would have been 12 : 23- As an example of the method of coincidences for nearly equal times of swing, we may take the accurate determina- tion of g by the aid of Rater's pendulum. Consider the vibration of a body in the form of a long metal rod, fitted with a spherical ball which can slide along it and be secured in any desired position. A knife-edge is fitted to one end of the rod in such a way that when it rests on a pair of horizontal plates the rod hangs vertically and can oscillate about the knife-edge. The other end of the rod is pro- longed to form a wire pointer. The rod is placed in front of the pendulum of a clock, and a telescope adjusted to view the two, which are arranged in such a way that when hanging vertically the rod-pendulum is exactly in front of^ and hides, some definite and easily recognised mark on the clock pendulum. To obtain such a mark a small silvered bead may be permanently attached to the clock pendulum, and a lamp arranged in such a position that the light from the bead as it passes through the lowest point of its path it reflected into the telescope. Thus at each transit of the bead an observer sees a bright flash of light If, however, at the same moment the other pendulum is also at the lowest point of its swing, this flash is cut off and does not appear in the field of view of the telescope. In practice it may, of course, happen that the flash is not entirely eclipsed at any transit As the observer watches he will see it grow dimmer and then again become brighter ; the transit at which the brightness was least will Ch. VI. § aaj Mechanics of Solids. 1 57 give the nearest approach to coincidence. Or, again, the flash may be obscured for more than one transit ; by taking the mean of the times for which this happens the time of co- incidence may be found. Now let us suppose the clock pendulum to be vibrating rather the more rapidly of the two. Watch the flash through the telescope, and, after noting the time, count seconds in time with the ticks of the clock until coincidence occurs. Write this time down. Do the same for a second, third, and fourth coincidence for motion in the same direction as the first, and thus find the interval, by taking the mean interval for several observations, between two coincidences. Suppose that during this interval n swings of the clock pendulum have occurred. In each swing the clock pendulum has gained a little on the other, and when it has completed n swings the other pendulum has made one less. So that the time of if — i swings of the latter pendulum is equal to that of n swings of the clock. Thus if / be the time of swing of the clock, t that of the other pendulum, (« — i) T = « /, T= /. « — I If, on the other hand, the other pendulum is going the more rapidly of the two, we should get t = « //(«+ 1).* The time of swing of the clock pendulum is obtained from astronomical observations. Direct observation is usually sufficient to determine whether the clock pendulum or the other is going at the quicker rate. The method * If we denote by T and / the times of half-vibrations of the pen- dnlams, and consider only transits in one direction, when a coinci* dence oc:urs again the one pendulum has lost or gained one whole or two hall vibrations, and thus we get nt - (fi + 2) T, or (#f — 3) T, as the case may be, and in this case T « /, or /, instead of « + 2 n-2, as above, n being the number of half-swings of the clock in the Interval between coincidences. 158 Practical Physics. (Ch. vi. § «x gives the time of the free pendulum, if it does not differ greatly from that of the clock, with great accuracy. Thus, suppose there is coincidence once in every 300 swings of the clock, then we have — 3~/i 1 =/x 1003344 . . . . (l) Whereas if we had found that the coincidence occurred every 301 swings, we should have obtained the value T=/x 1003333 . • . . (2) Thus the error made by a mistake of one in the number of swings between two coincidences is only 'ooooii of the time of swing. It must be remembered in the above that / is the time P^^ ^ of a complete oscillation, i.e. the interval between two transits in the same direction. For the application of the method in Rater's iAa pendulum the brass rod of the pendulum is fitted I with knife-edges (a, b, fig. x) at each end. In ^j. general the times of swing about the two knife- edges will be different ; but by adjusting the sliding weight E they can be made equal. If, when this is the case, h is the distance between the knife- • c edges, and t the time of swing, we can make use of the formula g Now if the pendulum be so constructed that T is very nearly one second, its value can be found with great accuracy by the method of coincidences, while the value of h can easily be determined by reading-microscopes (§ 5). The above formula thus gives us an accurate value for g. In practice it is not easy to adjust the pendulum so that the Ch. VI. § 2a] Mechanics of Solids. 1 59 times of swing from the two knife-edges are exactly equal ; if they differ slightly, a small correction to the above for- mula is required. It is shewn in books on dynamics that if /| l<i are the distances between the knife-edges and the centre of gravity, T| T2 the times of swing about the two knife-edges, then Now /j +/2 IS equal to h^ and can be found accurately ; the position of the centre of gravity may be roughly deter- mined by balancing the pendulum, and thus approximate values obtained, l^ and l^. If Tj is nearly equal to Tj, these approximate values are sufficient, for the last term -) — * will be very small, unless /, is too nearly equal to A^ Experiments, (i) Determine by observations on a simple pendulum the value of ^. (2) Compare the times of oscillation of the two pendulums. Enter results thus : — (i) Approximate value of /(from 31 transits) 4*15 sec Corrected value from an interval of 4 min. 8*66 sec . . . • • 4*I44 » Length of suspending wire . . .421-2 cm. Radius of bob •••••• 4'5 >, Value of/ 4257 „ (sec)' (2) Ratio of times from rough observations . 2-i Interval between coincidences twelve complete oscillations of the longer (the shorter pendulum gaining on the other). . Ratio of times . . , 2-083. (3) Determine by the method of coincidences the timci of i6o Practical Physics. [Ch. VI, § aa vibration of the given pendulum when supported from the two knife-edges in turn. Arrange the sliding weight so as to make these times more nearly equal, and hence determine the value of ^. Enter the results thus : — Pendulum adjusted so that the time between two coind- dences was approximately 24 seconds, the coincidences in one direction only being observed. The period of the clock pen- dulum is 2 seconds. An approximate value oin is therefore 12. Pendulum erect Coincidences observed at 15 m. 55 s., 19 m. 10 s. . , . . 30 m. 9 s. There have been 8 coincidences in the first interval, and «=I2*2. Using this value, we find there have been 35 coincidences in the second interval, and n^ 12-2. Pendulum inverted. Coincidences at 35 m. 35s., 38 m. 48 s., • . . . 49 m. 45 s. From the first two a more approximate value of n is 12*05, while from the first and third we obtain the more accurate value ff » 12*145. From these we find — Tj = i '8484, T,= 1-8478. Also A + ^9 •= ^4'88 cm. A -A -55-46 „ — » 'OIOO53 + -000003, o ^« 981-48 (cm.) sec-'. 21. A.twood'8 Machine. Two equal weights each of mass m are hung by a fine string over a pulley. A third weight of mass r is allowed to ride on one of these two, thus causing it to descend. After it has fallen through a measured distance, r is removed by means of a ring, through which the weight carrying it can pass» while R cannot. The time which it takes for the weights to fall through this measured distance is noted. Cn, VI. § 21.] Medianics of Solids, I6i After R has been removed, the other weights continue moving, and the time they take to pass over another measured distance is observed. Now, let us suppose that the height through which r ^s before being removed is a centimetres and that the time it takes in falling is / seconds. Let the space through which the weight continues to move downwards be c centimetres, and the time taken t^ seconds. Then, if for the present we neglect the friction and momentum of the pulley which carries the weights, the mass moved is 2M+R; and the force producing motion is the weight of the mass r; and hence, if / be the acceleration (tM + R)/=R^; whence / _ R^ "Ui 2M + R Also, since R descends through a space a in / seconds, a^\ft^\ and if v be the velocity acquired by the weights Fio. 13. at the time when r is removed, v^i and Thus, so long as the weights and rider r remain the same, we must have a proportional to the square of /. The distance a is easily measured by means of a measuring tape. Thus, let D (fig. 13) be the ring by which r is removed, and let a and b be the weights in their initial position. Lower the tape from d to the ground, and note the division with which the top of a coin- cides. Then release the string and allow the weight to fall, noting the interval /. Next, by pulling the string, raise the weight b until its top comes level with the ring, and note the division -(] aD %w ftM isrt of the tape opposite to which a stands. M 1 62 Practical Physics, [Ch. VI. § 21. The difference between these two readings gives the distance a. Thus, in the figure, a stands at 12 ft 8 in., when b comes to b' just passing the ring d, a has arrived at a', and the reading is 8 ft. 4 in. Thus a=i2ft. 8in. — 8ft 4 in.=4 ft 4 in.=i32*o8cnL We must now shew how the time / may be conveniently measured. This may be done by means of a metronome, a dock- work apparatus, which by adjusting a movable weight can be made to tick any required number of times — ^within certain limits — ^in a second. Adjust the weight so that the rate of ticking is as rapid as can conveniently be observed, and count the number of ticks in the time of fall It will be an advantage if the metronome can be so adjusted that this shall be a whole number. Then determine the number of ticks per second, either by the graduations of the metronome or by taking it to a clock and counting the ticks in a known interval, and thus express the time of fall in seconds. If a metronome is not obtainable, fairly accurate results tnay be obtained by allowing mercury to flow from a small nozzle through a hole in the bottom of a large flat dish, and catching in a weighed beaker, and then weighing the mercury which flows out while the weight is fidling. The weight of mercury which flows out in a known interval of time is also observed, and by a comparison of the two weights the time required is determined. The time / should be observed at least twice for the same fisdl a. Now make the same observations with a different fall, a^ suppose, and shew that the law that the space traversed varies as the square of the time is true.' ■ If the apparatus can be arranged so that the distance a can be raried, more accurate results may be obtained by determining the value Ch. VL § 21.] Mechanics of Solids, 163 Now, let the weight b, after falling through the distance n,. deposit R upon the ring d, and observe the time required by the weights a, b to pass over a further distance c \ let it be /i seconds. The weights move over the space c with uniform velocity v \ thus /j, the time of fall, is inversely propor- tional to V. Now, V is the velocity acquired by falling through the distance a ; thus v b proportional to the square root of a. Thus, /i should be inversely proportional to the square root of tf, or /j* proportional to if a. Thus, tf /,• should be constant, and equal to ^/2/ Observe the value of /| for various values of a^ and shew that a /|* is constant. From the last observations we can calculate the value of g; the acceleration due to gravity. For if / be the acceleration produced by the weight of the mass r, •^ 2M + R' P» = 2/tf, C^Vfil .\ ^=Z;«/l»=2/flV 2a /i* 2M + r' 2M + R 2_ ^ 2a /,>• X M and R are the nimiber of grammes in the weights used. We have neglected the effect of the momentum pro- duced in the pulley and of friction. We can allow for the former in the following manner : — of a, for which the time / is an exact multiple of the period of the clock or metronome. M 2 164 Practiced Physics. [Ch. VL § 21. It can be shewn theoretically that its effect is practically to increase the mass moved without altering the force tending to produce motion. Thus we should include in the mass moved a quantity w, which we can calculate by theory, or better determine by experiment. Thus, if/ as before be the acceleration, 2M + R + W 2at^ Repeat the observations, using the same value of c and II, but altering the rider to r' ; /, will be changed to /,', and the acceleration will be/' where ■^ 2M + R' + W 2a//>* /, ~/r'/i'«=2M4r'+w. ^ R/l*=2M + R + W. C^ ^(R/.«-R'//«)=R-R'. But Hence and * 2fl(R/,«-R'//*)' To eliminate the effect of friction we may determine experimentally the least mass which we must attach to the weight B in order just to start the apparatus. Let this be F grammes. Then, if we assume the friction effect to be constant throughout the experiment, the part of R which is effective in producing acceleration is R- f ; we must there- fore substitute r— f for r throughout It is probably not true that the frictional effect is the same throughout; the apparatus is, however, so constructed Cii. VI. § 21.] Mechanics of Solids. 165 that it is very small^ and a variation from uniformity is un- important The string by which the weights are hung is generally thin ; be careful therefore lest it break. Experitnents. (i) Shew from three observations that the space through which a mass falls in a given time is proportional to the square of the time. (2) Shew with the above notation from three observations that at? is a constant. .(3) Determine the value of g^ using two or three different masses as riders. (4) Obtain from your results with two of these riders a value for ^ corrected for the inertia of the pulley. (5) Correct your result frurther for the friction of the pulley. Enter results as below: — Value of < % Exp, I. Value of i f Ratio ^ (1) 400 cm. (2) 300 „ (3) 200 „ # 7*5 sec 6-5 » 5-4 n 7-1 7-1 6-9 Value of a (i) 400 cm. (2) 300 „ (3) 200 „ Exp, 2, Value of *» 4'3 sec 4*9 i> 61 „ 739 720 744 Exp,i, a «■ i M - : (1) R - (2) R' - (3) R"- ipo cm. joogm. 10 „ 8 n 6 n c - 45c - 4-3 -4*5 - 5-3 i cm. sec. Values of ^ respectively — 945 942 946 1 66 Practical Physics, [Ch. VL § d. D. The Fly-wheel. The kinetic energy of a particle of mass m moving with velocity v\&\mi^. If the particle be describing a circle of radius r with angular velocity a;, then tr = r w, and the kinetic energy becomes \m f^ nfi. The momentum of the particle v&mv^Qxmr w. The moment of this momentum about the centre, or the angular momentum of the particle, is m rv, or mr^ w. II the particle form part of a rigid body rotating about a fixed axis, then or, the angular velocity, is the same for all the particles. Thus the whole angular momentum of the rotating body is w S (m r^), or K <tf ; and the whole kinetic eneigy is ^ w* S (m^), or ^ K w* ; M being the mass of the body, and K its moment of inertia about the axis.^ Let h be the distance of its centre of gravity from the axis. * Mommt of Inefiia, —The moment of inertia of a body about a given axis may be defined physically as follows : — If a body oscillate about an axis under the action of forces which, when the boidy is dis- placed from its position of equilibrium through an angle 0, produce a couple tending to bring it back again, whose moment about the axis of rotation is ftB, then the time of a complete oscillation of the body about that axis will be given by the formula /=2ir V^ Hhere K is a < constant ' which depends upon the mass and configuration of the oscillating body, and is called the moment of inertia of S\e body about the axis of rotation. It is shewn in works on Rigid Dynamics that the relation between the moment of inertia K and the mass and configuration of the body is arrived at thus:— K is equivalent to the sum of the products of every small elementary mass, into which the body may be supposed divided, into the square of its distance from the axis about which the moment of inertia is required, or in analytical language K » Smr* (Routh's < Rigid Dynamics,* chap. iii). The following are the principal propositions which follow from this relation (Ronth's ' Rigid Dynamics,' chap, i) : — (i) The moment of inertia of a body about any axis is equal to the sum of the moments of inertia of its separate parts about the nme axis. (2) The moment of inertia of a body about any axb is equal to the moment of inertia of the boay atx)ut a paraUel axis through the centre of Ch. VI. § D.J Mechanics of Solids, 167 Again denote by i) the rate of cliange of velocity, i.e. the acceleration in the direction of motion, and by o> the angular acceleration- Then v=z I ik Let F be the force acting in the direction of motion. Then, since rate of change of momentum is equal to the impressed force, Now among the forces f we must reckon those which arise from the mutual reactions of the particles of the body. But if p, Q be any two particles of the body, and if Q act on p with a certain force f in any direction, then since action and reaction are equal and opposite ; p acts on q with a force — f. Thus the mutual reactions contribute nothing to the product 2 (r f), which therefore measures the moment about the axis of the impressed forces. gravity together with the moment of inertia of a mass equal to the mass of the b<Kly, supposed collected at its centre of gravity, about the original axis. (3) The moment of inertia of a sphere of mass M and radius a about a diameter is M|a'. (4) The moment of inertia of a right solid parallelepiped, mass M, whose edges are 2^l, 2^, 2c, about an axis through its centre perpen- dicular to the plane containing the edges b and c is "-3- (5) The moment of inertia of a solid cylinder mass M and radius f about its axis of figure is about 9L. axis through its centre perpendicular to the length of the cylinder. '(i*^ where 2/ is the length of the cylinder. It is evident from the fact that in calculating the moment of inertia the mass of each element is multiplied by the square of iu distance from the axis, the moment of inertia will in general be different for different distributions of the same mass with reference to the axU. i68 Practical Physia. [Cn. Vl. $ d. Thus we have as the equation of motion of a body about an axis the following : — Moment of impressed forces = Moment of inertia multiplied by angular acceleration ^ Rate of change of angular momentum. This statement, then, is the expression of the second law of motion applied to a body rotating about an axis.' The principle of the conservation of energy also tells us that the increase in the kinetic energy of the body is equal to the work done by the impressed forces. Thus, if we suppose the body to start from rest we have the result ^ K u* = Work done by the impressed forces. This result can readily be deduced from the former. We may exemplify the above by considering the motion Fio. n, of a flywheel mounted so as to turn on a horizontal axis B c (fig. xi) without much friction. A loop at the end of a piece of string is passed round a pin d on the axle, and a weight g attached to the other end of the string wound up hy turning the wheel. When the weight ia released the string is unwound off the axle, thus turning the wheel as the weight descends ; when the string is completely unwound the loop is released from the peg and the weight falls freely ; the wheel con- tinues to rotate until stopped by friction. ■ The above stalement has been deduced >lleT the usual method fram the second Inw of motion as applied (a linear motion. It majr be noticed that the science of dynamics may be based upon it ai a rundamentai law analogous to the second law al motion. Substituting couple for force, angular velocity and acceleration for linear velocity and acceleration, moment of ioertia for mass, we gel for rotation about an axis a series of propositions exactly conespondmg to those foi linear motion. Both systems give, of course, the same expression for kinetl: energy when the moment of inertia i< expie^scil is S n r>. Ch. VI. § D.] Mechanics of Solids, 169 Now it is shewn by the results of experiments that when the wheel is started the friction remains nearly the same, and is independent of the velocity, so that the work done by the friction in each turn is the same at whatever rate the wheel is moving. Let this work be f; let ui be the angular velocity of the wheel and k its moment of inertia. The first step is to find f. Wind up the weight, then release it, and after the string has fallen off let the wheel make n complete turns before coming to test ; let w be the angular velocity of the wheel at the moment the string falls off. The kinetic energy of the wheel at that moment was \ k b^^ and by the time the wheel stops this has been used in doing work against the friction. The amount of work so done is f . n, and hence F N = ^ K w^ If, then, we know k, and can observe n^, we can find the work done against fiiction. To find b», a strip of metallic paper is fastened on to the rim of the wheel by india-rubber bands or otherwise, and a tuning-fork, carrying a light metallic style and vibrating in a horizontal plane, is arranged so that the style can be readily brought into contact with the paper by pushing the stand of the tuning-fork against fixed stops on the table. The tuning-fork is set vibrating, and as the weight falls off it is moved so that the style just touches the paper. A wave-curve is thus drawn on the paper, and, as in § B, the wave-length gives the distance traversed by a point on the rim of the wheel during one vibration of the fork. Multiply- ing this by the number of vibrations per second, we get the velocity «, say, of the rim at that moment, and dividing this by a, the radius of the wheel, we have the required angular velocity ii>. Again, suppose we consider the motion while«the weight is still on the string, after the wheel has made n revolutions J 70 Practical Physics. (Ch. VL § ix from the start, and the weight has descended a distance % cm. Let m be the mass of the weight. Let r be the radius of the axle, v the velocity of the falling weight This is, of course, the same as that of a point on the axle. Since a length 2 7rr of string is imwoimd at each turn, the weight descends through this distance in one turn, and since z is the fall in n turns, » = 2«7rr. Also ZF = r w. Now in descending a distance % the weight has lost an amount of potential energy mg% \ this has been used (i) in giving kinetic energy to the wheel ; (2) in giving kinetic energy to the weight ; (3) in overcoming the friction. Thus a « 71 rmg = ^ (»i f* + k) w* + F «. • • We have already seen how to find f. If we substitute its value in this equation we can determine <•», and then verify the result by finding the same experimentally. The quantity k includes the moment of inertia of the axle as well as that of the wheel If m be the mass of the wheel, treated as a uniform circular disc, m' that of the axle, then The apparatus can usually be arranged so that the last term is very small The following method will give us the friction without an accurate knowledge of a;. After the weight has fallen off let the wheel continue to run, and suppose it make vl turns before coming to rest. Then ^ K w' = F If ', /. mgz = i»i2^ + F(« + «'). This result is obvious, for, since the wheel has come to rest Ch. VI. § d.) Mechanics of Solids, 171 again, the potential energy of the fallen weight has been used in producing kinetic energy \mv^m that weight, and in doing work f (« + n') against friction in (« + Ji') turns of the axle. Now we can usually arrange the experiment so that \fnv^ \s very small compared with mgz^ and when this is the case Even if we cannot entirely neglect the kinetic energy of the weight, an approximate value of a;, and therefore of v, will enable us to calculate the term i^mv^ with sufficient accuracy. Experiment. — Find the angular velocity generated by the effect of the given couples in measured intervals of time, and deduce the moment of interia of the fly-wheel. £. Pendulum of any shape. A simple pendulum consists of a mass m attached at one end of a string or weightless rod of length /, and allowed to vibrate about the other end If such a pendulum be dis- placed a distance x measured along its path from its equilibrium position, then we know (see Maxwell, * Matter and Motion,' art cxix.) that it has potential energy mea- sured by mgx^Jzl, Moreover, if a mass m execute simple harmonic vibra- tions {n per second), the potential energy at a distance x is 2mif^fi^x\ and these two expressions for the energy must* be equal Thus : — 2tV / A similar method of reasoning may be applied to the case in which the pendulum is not a simple one, but con- tists of a rigid body vibrating about a horizontal axis. For 172 Practical Physics. [Ch. VL § «- take the plane of the paper as the plane of motion ; let the axis cut this plane in o ; then the forces acting on the body are its weight and the reaction through o. Now as the body oscillates, the point o, through which this reaction acts, remains fixed, and no work is done on the body by the reaction. The changes in the potential energy then depend only on the weight of the body which acts vertically through its centre of gravity and on the position of the centre of gravity. The potential energy is the same as it would be if the whole mass were concentrated at the centre of gravity. Thus if M be the whole mass of the body, h the dis- tance of its centre of gravity below o, and the angle through which a line through o is at any moment dis- placed, the value of the potential energy is for this is the expression we have found for the potential energy of a mass m oscillating at a distance h below a fixed point But taking the second expression for the potential energy given above, since the number n of vibrations per second is the same for each particle, we have Total potential energy=2(2»i7r2«2*a)=27r2«^2:(/;//»^a) where m^* expresses the result of finding the value, ^(mP), for all points. Mk^ is clearly the moment of inertia of the body, and k is known as its radius of gyration. We may Jipmetimes con- veniently denote the product m>^^ by a single symbol k.' Thus we have •' 2;rV -k iiV IP' ' Sec footnote, p. i66. Ch. VI. § E.J Mechanics of Solids. 173 FicxIL Now the value of k can be calculated* for bodies of certain definite forms (see p. 167). If K be known, we can use a rigid pendulum to cal- culate gi liy on the other handy K be not known, we can use the above result to find it, provided we know^ and can find n and m h. The following measure- ments will give us m h. Attach a fine string to some point p of the pendulum (fig. xii), pass it over a good pulley L, and fasten a mass m' to the end. Then the tension of the string is u'g. Let o N be perpendicular to the direction of the string. Let Q be the angle between the displaced position of o o and the vertical. Taking moments about o, we have Fig. xHU u!.g. ON = M.^.OGsm6 = ug h sin Q ; /• li ^ = m'. o n cosec 6. In practice it would be sim- plest to arrange the pulley so that the string is horizontal ; then o N is vertical, and the equation to find k becomes, = — L_-_ m' , ^. o N cosec 0, To find cosec 6 mark with a plumb-line the vertical line through o before the body is displaced. This becomes 1 74 Practical Physics. [Ch. VI. § t. o G. Let the direction of the string, supposed horizontal, cut the displaced position of this line in Q (fig. xiii), then cosec 6 = q/q n, and 4 TT* «* ^ Q N The following is another method of finding k. Attach to the given body, so as to vibrate about the same axis, another body whose moment of inertia about the axis can be calculated. Let k' be this moment of inertia, m' the mass of the body, and h' the distance from o of its centre of gravity. Observe the time of vibration ; let it be i/«', then we have «« = I M^^ 4 7r> K ' " ""4T« K + K' From these two equations we can eliminate m h^ and if m', h!^ and k' are known, can find K. Experiment. — A rectangular bar of iron is made to vibrate about a knife-edge near one end at right angles to its length. Find the value of its moment of inertia, (i) by the first method ; (2) by attaching to one end a sphere of lead. F. Ballistio Pendulum. Heasurement of Homent of Homentum and of Homantum. If a moving body {e.g.y a moving iron ball or a hammer head) comes in contact with a heavy body (called for brevity a ballistic pendulum) having a definite position of equili- brium, but firee to rotate about a horizontal axis, then (i) the momentum of the moving body is changed by the impact, and the change of momentum measures the impulse of the Ch. VI. s f . J Mechanics of Solids, 1 7 5 blow delivered by the one body and received by the other ; (2) the pendulum starts from its position of equilibrium with an angular velocity such that the moment of its momentum about the axis of rotation is equal to the moment of the impulse about the same axis, or, to put the case in more general terms, if the pendulum is already moving when the blow takes place, the change of moment of momentum about the axis of rotation is equal to the moment of the impulse about the axis. These two state- ments are derived directly from Newton's laws of motion (see p. 168). The pendulum will reach a position of instan- taneous rest at the extremity of its first swing when the kinetic energy of its motion has been converted into the potential energy due to the lifting of the centre of mass of the pendulum against the forces of gravitation, allowance being made for work spent in overcoming the forces due to friction with the air and at the axis of rotation, the effects of which may usually be neglected. It will then swing back again and oscillate with gradually diminishing ampli- tude. In computing the change of momentum of the impinge mg body we should require to know its velocity before and after impact as well as its mass. If the material of the surface be plastic, as lead or putty is, and the pendulum be, comparatively speaking, very heavy, the impinging body will be simply stopped by the blow, and the mea- sure of the impulse then depends merely upon its initial velocity. We could clearly compute the moment of the impulse if we could measure the angular velocity a; communicated to the pendulum provided we knew its moment of inertia K \ for the moment of momentum, to which the moment of the impulse is equivalent, is k cii ; instead of measuring directly the angular velocity, we may deduce it from obser- vations which are easier in practice — viz., the amplitude of £76 Practical PJ^ysics. (Ch. VI. § F, the first swing after receiving the blow, and the time of vibration of the pendulum. We shall describe a form of apparatus suitable for use in a laboratory, in which the principles above indicated can be practically applied to the measurement of the moment of an impulse, and consequently to the calculation of the change of momentum produced by a blow. A somewhat similar form of apparatus has long been known under the name of the ballistic pendulum, and has been used to measure the initial momentum of a rifle bullet, and an apparatus based upon precisely similar djmamical principles is regularly used as a 'ballistic galvanometer needle' to measure transient electric currents. (See chap. xxL) The apparatus is represented in fig. xiv. a a' is a long beam, tightly gripped in the two halves of a groove cut in a Fic. xiv. d n » a 3t fmn 1 frt i«i i i D w V- 3 D pair of thick boards, which are shaped into segments of circles somewhat greater than a semicircle, and which form when screwed together a substantial block b. The whole can swing on a knife-edge, fixed so that the axis of rotation coincides with the common axis of the semicircles. The top face of the beam passes through the axis, and the beam Ch. VI. § r.J Meclianics of Solids, 177 is graduated on each side, from the line where the knife- edge meets it Two weights of measured and equal mass are hung by knife-edge attachments (or simply by wire loops) from corresponding graduations on the two sides of the centre. These are used to alter the moment of inertia of the pendulum without altering its total mass or the position of its centre of gravity, so that in dealing with its oscillations the quantities denoted by m and h (p. 166) may be regarded as the same for all positions of the movable weights. At the top of the block b is firmly fixed a rect- angular block of wood R to receive a horizontal blow at a marked point To administer a horizontal blow a pendulum-bob, supported in the proper position by a V-shaped suspension can be used. The advantage of this arrangement is that the momentum of the bob can be cal- culated frt>m its mass and its initial displacement A vertical blow can be given (by a hammer or a falling mass) upon a stud driven into the horizontal face of the block B. The knife-edge must be supported in a horizontal position by blocks on each side, the edge lying in a shallow V-groove. The whole pendulum is then symmetrical about the vertical plane through the middle of the beam and the vertical plane through the knife-edge. The angular deflexion produced by a blow can be found roughly by graduations on the circular edge of the block B, or, more accurately, by reading, as with a mirror galvanometer, the displacement of the image of a scale viewed in a mirror m attached to the extremity of the knife- edge. A pin placed at the same distance behind the mirror as the image of the scale, and an opera-glass, will enable the experimenter to dispense with the darkening of the room. To facilitate adjustments the apparatus above described should be completed by the addition of arrangements 178 Practical Physics, [Ch. VI. § r. p and i) corresponding respectively to the flag and inertia- bob of a balance, in order that the centre of gravity of the whole may be slightly moved horizontally or vertically, as may be found necessary. The experiments which may be performed with the ballistic pendulum are as follows : — (tf) The observation of the time of vibration with the movable weights in two different positions, and the calcula- tion from the observations of the moment of inertia of the pendulum. ip) The observation by means of the mirror and scale of the amplitude of the first swing when a horizontal blow is struck by a pendulum-bob pulled aside through a measured vertical height, and the calculation of the momentum of the impinging bob. The variations of the effect caused by the ballistic pendulum not being quite at rest and by an altera- tion of the material upon which the blow is delivered can also be observed. The observations may be repeated with the movable weights at various distances. (r) The observation of the deflexion due to the blow of a hammer or other impulse of unknown magnitude, with a view to its measurement {d) Observation of the permanent deflexion due to a force of known moment about the axis of rotation. The theory of the working of the apparatus is as follows : — (a) For the determination of the moment of inertia k, when the weights, each of mass w, are at a distance / from the centre. If r is the observed time of vibration, k^ the moment of inertia of the pendulum without any movable weights, M the mass of the pendulum, and h the distance of its centre of gravity below the axis, rsss 2Try/ VijUgh = ax>/(Ko + 2W/*)/M^A . . (i) Let the weiglits be moved to a distance /', let the corre- Ch. VI. § r.j Mechanics of Solids, 179 sponding time of vibration be r'. Neither m, nor h^ nor g is altered, hence whence K = aWr2(/'«-/«)/(r^a-r«) , . . (2) {b) and (^) For the calculation of the moment of the impulse from, the amphtude of the first swing, let or be the initial angular velocity due to the impulse, then k ci> is equal to the moment of the impulse. The initial kinetic energy is ^ K w^ When the penduliun is in equilibrium, the distance of its centre of mass below the point of suspension is repre- sented by ^ ; when it has been deflected through an angle a, the centre of mass has been raised through a height ^(i— cos a), /.^. 2^ sin* -, and the work done in raising 2 it is 2MgJk sin*^, if a represents the amplitude of the 2 first swing. The work done in the raising is the equivalent of the kinetic energy which has disappeared (neglecting the losses on account of friction). Hence we have i^Kuf^ = 2MgJk sin' -, 2 and, making use of equation (i), we get « = 4rsin-/r (3) 2 and the moment of the impulse is 47r k sin - /r. If the ver- 2 tical distance of the point at which the blow is delivered above the knife-edge is ^, we get the impulsive change of momentum of the moving body to be 47r k sin -/r^. 2 (d) The permanent deflexion produced by a known mass, w, hung fi-om the beam at distance X, is merely the result of using the pendulum as if it were a balance, and N 2 I So Practical Physics. (Ch. VL § ». the theory is that of the balance. Hence, if 6 is the de- flexion produced, t9xX = M^ tan % whence M^ = w\ cot 0. This equation enables us to determine the moment of inertia k from equation (i) without an observation of the time of vibration for an altered position of the movable weights. (Compare the corresponding use of the deflexion produced by a steady current in the ballistic galvanometer. Chap, xxi.) We need only add a few practical details. Measurement of Times of Oscillation. — There should be considerable distance between the two positions of the weights. On moving the weights to a new position, the position of equilibrium of the pendulum, as read by the mirror and scale, must be adjusted to be the same as before. The times may be taken with sufficient accuracy by timing, say, fifty vibrations with a stop-watch. Measurement of First Swing, — ^To secure that the blow is horizontal the bob should be arranged to hang freely, just touching the block r when the pendulum is at rest It is important that the pendulum should be quite at rest when the blow is delivered, and the position of equilibrium read before each observation. One observer should watch the image of the scale, while another lets the bob go. If the motion is too rapid for the scale reading to be satis&ctorily taken, a piece of black thread may be tied round the scale, and gradually adjusted until its reflected image is just reached on repeating the observation. The scale reading of the thread can then be taken at leisure, and the difierences for difierent successive observations can be esti- mated. The ratio of the scale reading of the deflexion divided by the distance of the scale from the mirror (both ex* Ch. VI. { F J Mechanics of Solids, l8i pressed in the same measure) is the tangent of twice the angle of deflexion, and may for small angles be taken to be equal to the sine of the double angle. The sine of the half angle may accordingly be taken as one quarter of the latia The angular deflexion ought to be small in any case, as the law of isochronous vibrations does not apply with sufficient accuracy when the oscillations of a pendulum are of considerable amplitude. Observations may be recorded in the following form : — Distance of scale from mirror . Mass of sliding weights, each . Distance of sliding weights from centre Corresponding time of vibration First swing after im- pulse 1 1 m. " 27*94 cm. 4ohalf-vibrations in 49 sees. 330-230 a 100 mm. . 6o'3 cm. . 2050 grammes 22 in. » 55*88 cm. 30 half- vibrations in 6275 sees. 333 - 280 -53 mm. cm.' sec. cm.' sec. From time observations, Kq" 1'6o2 X lo** gm. cm.* Kq + Kji - 4'8o8 X lo* gm. cm.', Kq + K^- 1-442 X 10' gm. cm.' From first deflexion observation, Moment of impulse « i -022 x 10^ gm. From second deflexion observation, Moment of impulse = i*oi6 « 10" gm. _ , , n cm.* Mean moment of impulse - i'Oi9 x 10* gm. — sec Vertical distance of point of impact from axis- 1 78 cm. Weight of bob, 320 gms. Vertical height of fall required to generate the impulse = {1-019 X lo^jij'S X 32o}'/98i X 2 = 16-3 cm. It will be evident that the same apparatus can be used to illustrate the logarithmic decrement of oscillations and some other interesting dynamical questions which we have not space to discuss. 1 8;? Practical Physics. (Cil VI. § g. G. Fanicnlar Polygon. OrapMo Method of Comparing Forces. If three forces acting at a point are in equilibrium, they can be represented in magnitude and direction by the three sides of a triangle taken in order. This is the proposition known as the ' triangle of forces.' If the directions of the three forces are given, and if a triangle be constructed with its three sides respectively parallel to those directions, the magnitudes of the forces are proportional to the lengths of the respective sides, and the forces can be compared by measuring and comparing the lengths. We shall illustrate this method of comparing forces by applying it to the follow- ing special case, an example of what is well known as the funicular polygon. WeightSy W|, Wa, W3, 6^^., the mass of one of which^ w^, is known y are hung from separate points ^ Aj, A2, A3, of a string. The ends of the string are made fast to two fixed points^ A and b. JFind the mass of Wj and of W3, and the tensions of the several portions of the string. We must first draw a scale diagram, a Aj Ag A3 b (fig. xv), in which the directions of the forces, as indi- cated by the strings, are correctly given. For this purpose mount a graduated straight-edge accurately horizontal above the higher of the two points a, b, and set out on paper a straight line to represent the horizontal edge. Then with a T-square and rule, or a plumb-line, measure the vertical distances a a, ^iA,, &c., of the respective points A, Ai, A2, A3, B, below the straight-edge, and read also the horizontal distances aa,, a^ a^t &c. Set out these horizontal and vertical distances in the diagram to any convenient scale, and join by straight lines the points repre- senting A and A„ A, and A2, &c., respectively. These join- ing lines, together with the vertical, shew the directions of all the forces acting at the several points. Ch. VI. § o.) Meclianics of Solids, 183 At each of the points A|, A2, A3 three forces act, namely, two tensions and a weight We next proceed to construct a triangle (fig. xvi) with its sides, taken in order, parallel to the three forces Wi, Xi, and t^, acting at Ai. First set out X|Xj vertical, and of a convenient length to represent accurately w,, the known weight, in magnitude. The line Xi X3 is parallel to Ai^i. Next, by a parallel ruler, or by a straight-edge and set square, draw a line XjO accurately parallel to a, a, and then from the point Xj draw XjO accurately parallel to a, Aq. The point o is the intersection of the last two lines, and a triangle o X| Xj has been drawn with its three sides, taken in order, parallel to w,, t,, and t, respectively, acting at the point a,. Their lengths there- fore represent those three forces in magnitude on the same icale as that on which x, Xj represents w,. 1 84 Practical Physics, [Ch. VLf a. Next produce the vertical XiXj, and through o draw 0X3 accurately parallel to A2 A3. Then, remembering that FicxvL ^^ force at A2, due to the tension, \& equal and opposite to the force at Ai, due to the tension of the same piece of the string, it is evident that o X3 X2 is a triangle with its sides representing in direction, and consequently in magni- 's tude also, on the same scale as before, the three forces acting at a 2. Similarly we can construct the triangle 0X4X3, and the sides of the triangles which are comprised in the figure ox,X4 represent in magnitude and direction all the forces acting. The magnitudes of these forces can then be compared by comparing the actual measured lengths of the respective lines. The lines to be measured in order to determine the magnitudes of the respective forces are indicated in fig. xvi. We have neglected the weight of the portions of the string itself, and in practice this is quite justifiable with good sized weights. It is, however, not a difiScult exten- sion of the same method to compare the tensions at differ- ent points of a heavy chain hanging between two points, and find the weight of unit length by observing the shape in which the chain hangs when a known mass is hung on one link. The tensions at the two ends of the string can be found by the use of a spring balance. Ch. VI. J Mechanics of Solids, 185 SUMMARY OF THE GENERAL THEORY OF ELASTICITY. The elastic properties of an isotropic homogeneous elastic body depend on two qualities of the body — ^viz. its compressibility and its rigidity. The compressibility de- termines the alteration in volume due to the action of external forces, the rigidity the alteration in form. Compressibility and Elasticity of Volume, Suppose we have a body whose volume is v, and that it is under a hydrostatic pressure p ; let the pressure be changed to P+/, and the volume in consequence to v— v. Then f^/v is the change in unit volume due to the increment of the pressure /, and vf^yp) is the change per unit volume due to unit increment of pressure. This is called the compressibility of the body, which may be defined as the ratio of the cubical compression per unit volume to the pressure producing it The reciprocal of the compressibility — viz. the value of wpjv — is the elas- ticity of volume. We shall denote it by k. Rigidity. Any alteration of external form or of volume in a body is accompanied by stresses and strains throughout the body. A stress which produces change of form only, without alteration of volume, b called a shearing stress. Imagine one plane in the body to be kept fixed while all parallel planes are moved in the same direction parallel to themselves through spaces which are proportional to their distances from the fixed plane ; the body is said to undergo a simple shear. Suppose further that this simple shear is produced by the action of a force on a plane parallel to the fixed plane, and uniformly distributed over it ; then the ratio of the force per unit of area to the shear produced is defined to be the rigidity of the body. 1 86 Practical Physics. [Ch. VI. Let T be the measure of the force acting on each unit of area of the plane, and suppose a plane at a distance a from the fixed plane is moved through a distance c\ then c\a is defined as the measure of the shear, and the rigidity of the body is Talc Let us call this iu It maybe slvftivn mathematically^ that, if a circular cylinder of radius r and length / be held with one end fixed, the couple required to turn the other end through an angle B'\%n -— 0. Modulus of Torsion. The couple required to twist one end of unit length of a wire through unit angle, the other end of the wire being kept fixed, is called the modulus of torsion of the wire. Hence if r be the modulus of torsion, the couple re- quired to twist one end of a length / through an angle 0, the other end being kept fixed, is r&lL Relation between Modulus of Torsion and Rigidity, We have given above two expressions for the couple required to twist one end of a length / of a wire of cir- cular section through an angle 0, the other end being kept fixed ; equating these two expressions we get for a wire of radius r, n ■^— M* wr* Young's Modulus, If an elastic string or wire of length / be stretched by a r—i weight w until its length is Z', it is found that -- — is constant /w for that wire, provided that the wire is not strained beyond the limits of perfect elasticity; that is, the weight w must be such that, when it is removed, the wire will recover its original length. If the cross section of the wire be of unit area, the ratio * See Poynting & Thomson's Physics. Ch. VI. § 22 ) Mechanics of Solids. 1 8; of the stretching force to the extension per unit length is called Young's Modulus, for the material of which the wire is composed, so that if the cross section of the wire be « sq. cm. and we denote Young's Modulus by e, we have Relation between Youngs Modulus and the Coefficients of Rigidity and Volume Elasticity, Wp can shew from the theory of elasticity (see Thomson, Ency. Brit, Art. ' Elasticity '), that if Ebe Young's Modulus, and hence ik-^-n' *=- «^ 3 (3«-E)* Thus, knowing b and n, we can find k. 22. Young's Modnltu. To determine Young's Modulus for copper, two pieces of copper wire seven or eight metres in length are hung from the same support One wire carries a scale of millimetres fixed to it so that the length of the scale is parallel to the wire* A vernier is fixed to the other wire,* by means of which the scale can be read to tenths of a millimetre. The wire is prolonged below the vernier, and a scale pan attached to it ; in this weights can be placed. The wire to which the millimetre scale is attached should also carry a weight to keep it straight Let us suppose that there is a weight of one kilogramme hanging from each wire. Measure by means of a measuring tape or a piece of ttring the distance between the points of suspension of the > We believe that we are indebted indirectly to the Laboratory of King's College, London, for this elegant method of reading the extension of a wire. 1 88 Practical Physics, [Ch. vi. § 2%. wires and the zero of the vernier. Let this distance be 718*5 centimetres. This gives the length of the wire. Read the vernier at the same time and let the reading be 2*56 centimetres. Now add 4 kilogrammes to the pan. The wire is stretched and the vernier descends relatively to the scale. Read the vernier again, and let the new reading be 279 centimetres. The length of wire to which the scale is attached is unaltered, and thus the increase in the length of the stretched wire is clearly the difference between these readings, or 0*23 centimetres, and this extension ha§ taken place on a length of 718*5 centimetres. Thus 4 kilogrammes stretches the wire by the difference between 2*79 centimetres and 2*56 centimetres. The elongation, therefore, is 0*23 centimetre, the extension per unit length is 0*23/718*5, and the ratio of the stretching force to the extension per unit length is l^i^— §, or 12500 kilogrammes approximately. •23 We require the value of Young's Modulus for the material of which the wire is composed. To find this we must divide the last result by the sectional area of the wire. If, as is usual, we take one centimetre as the unit of length, the area must be expressed in square centimetres. Thus, if the sectional area of the wire experimented on above be found to be 0*01 square centimetre (see § 3), the value of the modulus for copper is —1-?, or 1250000 kilogrammes per square centimetre. The modulus is clearly the weight which would double the length of a wire of unit area of section, could that be done without breaking it. Thus, it would require a weight of 1,250,000 kilo- Ch. VI. § 22.] Mechanics of Solids. 1 89 grammes to doable the length of a copper wire of one square centimetre section. The two wires in the experiment are suspended from the same support. Thus, any yielding in the support produced by putting on weights below or any change of temperature affects both wires equally. It is best to take the observations in the order given above, first with the additional weight on, then without it, for by that means we get rid of the effect of any permanent stretching produced by the weight The wire should not be loaded with more than half the weighfrequired to break it A copper wire of 0*01 sq. cm. section will break with a load of 60 kgs. Thus, a wire of o'oi sq. cm. section may be loaded up to 30 kgs. The load required to break the wire varies directly as the cross- section. To make a series of determinations, we should load the wire with less than half its breaking strain, and observe the length ; then take some weights off— say 4 or 5 kgs. if the wire be of about 0*01 sq. cm. section, and observe again ; then take off 4 or 5 kgs. more, and observe the length ; and so on, till all the weights are removed. The distance between the point of support and the zero of the millimetre scale, of course, remains the same through- out the experiment The differences between the readings of the vernier give the elongations produced by the corre- sponding weights. The cross-section of the wire may be determined by weighing a measured length, if we know, or can easily find, the specific gravity of the material of which the wire is made. For, if we divide the weight in grammes by the specific gravity, we get the volume in cubic centimetres, and dividing this by the length in centimetres, we have the area in square centimetres. It may more readily be found by the use of the wire- gauge (see § 3). igo Practical Physics. [Ch. vx § 22. Experiment — Determine the modulus of elasticity for thr material of the given wire Enter results thus : — Length of unstretched wire • • . 718*53 cm. Extension per kilogramme (mean of 4 ob- servations) "0575 » Cross-section "01 sq. cm Value of E 1,250,000 kilogrammes per. sq. cm. Modulus of Torsion of a Wirt. If the wire contain / units of length, and the end be twisted through a unit angle, each unit of length \& twisted through an angle i//, and the couple required to do this is r// where T is the modulus of torsion of the wire. The couple required to twist unit length through an angle ^ is r^, that required to t>vist a length / through an angle 9 is r6\L Suppose a mass, whose moment of inertia is k, is fixed rigidly to the wire, which is then twisted, the mass will oscillate, and if t^ sec. be the time of a complete oscillation, it can be shewn, in a manner similar to that of § e, that r.=2.^(£^. To find r, then, we require to measure /| and k. K can be calculated if the body be one of certain deter- minate shapes. If not, we may proceed thus : We can alter the moment of inertia of the system without altering the force tending to bring the body^ when displaced^ back to its position of equili- brium^ either (i) by suspending additional masses of known shape, whose moment of inertia about the axis of rotation can be calculated, or (2) by altering the configuration of the mass with reference to the axis of rotation. Suppose that in one of these two ways the moment of inertia is changed Ch. VI. § 22.] Mechanics of Solids, 191 (rom K to K+i^ where the change k in the moment of inertia can be calculated, although k cannot Observe the time of swing again. Let it be /,. Then *Thus Whence ,=2.y/^ +^y 4^»>^/ • • T= Thus T can be expressed in terms of the observed quan- tities /i, /) and /, and the quantity k which can be calculated. We proceed to give the experimental details of the application of this method of finding the modulus of torsion of a wire by observing the times of vibration, /|, /g, when the moments of inertia of the suspended mass are k and K+>& respectively. The change in the moment of inertia is produced on the plan numbered (2) above, by a very con- venient piece of apparatus devised by Maxwell, and described in his paper on the Viscosity of Gases. 33. To find the Modulus of Torsion of a Wire by Haxwell*8 Vibration Needle. The swinging body consists of a hollow cylindrical bar k B (fig. 14). Pic. 14. Sliding in this are four equal tubes which together just fill up the length of the bar ; two of these are empty, the other two are filled with lead. ^ CD is a brass piece screwed into the bar, and m is a plane mirror fastened to it with cement. At d is a screw, by means of which 192 Practical Physics. [Ch. VI. § 23. the bar is secured to the wire of which the modulus is re- quired. E F is a horizontal scale placed so as to be re- flected in the mirror m, and o h is a telescope adjusted to view the image of e f produced by the mirror. The eye- piece of the telescope is provided with cross-wires. The first adjustment necessary is to arrange the apparatus so that when the bar is at rest the central division of the scale, which should be placed just above the telescope, ap- pears, in the field of view of the telescope, to be nearly coincident with the vertical cross-wire. The mirror must be adjusted either by loosening the screw d and turning the bar round, or by turning the support which carries the wire, until when in the position of rest the plane of the mirror is very nearly at right angles to m g. When this is done, reduce the bar as nearly as possible to rest, and point the telescope towards the mirror. For this purpose focus the telescope on the mirror, move it until the mirror is seen in the centre of the field, and then fix it with a clamp. Alter the focus of the telescope so as to view an object at about the same distance behind the mirror as the scale is in front For the present this may be done quite roughly, by slightly pushing in the eye-piece. If the scale happen to be in adjustment, the image will be seen in the mirror. If this be not the case, move your head about behind the telescope until the scale is seen reflected in the mirror. Notice the position of yoiu: eye with reference to the tele- scope, and infer from this how the scale requires to be moved Thus, if your eye is above the telescope, the scale is too low, and vice versd. Move the scale in the direction required until it is in the field of view of the telescope, and fix it securely. There is another way of performing this adjustment, which may sometimes prove more rapid. Looking through Ch. VI- § 23.] Mec/ianics of Solids. 193 the telescope, move a lighted lamp or match about until a glimpse of it is caught reflected in the mirror. The position of the lamp at that moment shews you where the scale should be. (If the first method be adopted, it is easier to see the scale by going close up to the mirror until it comes into view, and then moving backwards to the telescope, still keeping it in sight) Suppose now the scale is seen reflected from the mirror; the central division of the scale will probably not coincide with the cross-wire. For many purposes this is unimportant. I^ however^ we wish to bring the two together we must notice what point on the scale will come opposite the cross-wire when the mirror is at rest,* and then turn the torsion head, which carries the wire in the right direction until the central division is brought into view. It may be impossible to make the adjustment in this manner ; in that case we must move the telescope and scale. Thus, if o be the central division of the scale and p the division which coincides with the cross-wire, the necessary adjustment will be made if we move the telescope and scale through half the distance o p, still keeping the former pointed to the mirror. It is sometimes necessary to set the scale at right angles to M o or M G. For this purpose measure with a string or tape the distances of e m and f m, and turn the scale round a vertical axis until these two are equal Then since o e=o p and M £=:M F, it is clear that o m and e f are at right angles, and the required adjustments are complete. To observe the Time of a Complete Vibtation, Twist the bar slightly from its position of rest, and let it vibrate. ' When the position on the scale of the ' turning-points ' of the needle can be read through the telescope, the position of equilibrium can be determined in exactly the same manner as m the case of the balance (see S laV 'lCji. »1 - z: ^v^ .icaeidof f*»a-JK •3C mstal at nc doss-waeoithe - a whidi the scBC ^ .1 'JIC SSBBCttnCtBC . uie cbiQOQiBBtr , 3S dtSCUDtd 4« ^\; h -laf >* Ch. VI. § 23.] Mechanics of Solids, 195 Of course, if we always count the same number of ticks there is no need to subtract the 3 sec. from the^ chrono- meter reading ; we are concerned only with the differences between the times of transit, and the 3 sec. affect\ all alike. We may thus observe /|, the time of vibration of the needle when the empty tubes are nearest the ends, the loaded tubes being in the middle ; and in the same manner we may observe /2> the time of the vibration of the needle when the positions of the heavy and light tubes have been interchanged. To find the Value of k^ the Increase in the Moment of Inertia, We know that the momentof inertia of a body about any axis is equal to its moment of inertia about a parallel axis through its centre of gravity, together with the moment of inertia of the whole mass collected at its centre of gravity about the given axis (p. 44). Thus, let m be the mass of a body whose moment of inertia about a certain axis is I ; let a be the distance of the centre of gravity from that axis, and I the moment oi inertia about a parallel axis through the centre of gravity. Then 1=1 +f«fl^ Moreover, the moment of inertia of a body is the sum of the moments of inertia of its parts (p. 44). Now, let ^1 be the mass of each of the heavy tubes, and « the distance of the centre of each of them from the axis round which the whole is twisting when in the first position. Let Ii be the moment of inertia of each of the heavy tubes about a parallel axis throtlgh its centre. Let m^ I3 have the same meaning for the empty tubes, and let b be the distance of the centre of each of these from the axis of rotation. Let I be the moment of inertia of the empty case. Then O 2 196 Practical Physics. ICh. VI. § 23. In the second position, a is the distance from the axis of rotation of the centre of each of the masses vi^ b of that of the masses nix. To find the moment of inertia of the whole, therefore, we require simply to interchange a and b in equation {\\ and this moment of inertia is k+>^. Thus, K + >^=I + 2li+2la + 2/«i^*f 2»l^*. • . . (2). from (i) and (2) >J=2(^'— tf*)(»«i— Wj). Thus, we do not need to know I, Ii or I3 to find k. Now the length of each of the tubes is one-fourth of that of the whole bar a B. Calling this ^, we have ^=8 ^~8' and >^=4^(»ii— Wj). To find m, and m^ we require merely to determine by Weighing the number of grammes which each contains. Our formula for r (p. 191) becomes and it only remains to measure /. This can be done by means of the beam compass or a measuring tape. We must, of course, measure from the point at which the upper end of the wire is attached, to the point at which it is clipped by the screw d. The wire it will be found fits into a socket at the top of the apparatus CD. Be careful when fixing it initially to push it as far as possible into the socket ; its position can then be recovered at any time. Unloose the screw d and dn^ the wire from above, up through the tube which supports it, and measure its length in the ordinary manner. The value of r thus obtained gives the modulus of torsion for the particular specimen of wire. If the rigidity of the material is required, we must make use of the addi- Ch. VL § 23.] Mechanics of Solids. 197 tional law of torsional elasticity that the torsional couple in wires of the same material, differing only in area of section, is proportional to the fourth power of the radius of the wire. To find the value of the rigidity of the material, the value of r must be divided by ^n-r* where r is the radius in centimetres (p. 186). Experiment. — Determine the rigidity of the given wire. Enter results thus : — A " 5*95 sec /, -975 sec. wi, « 3 5 1 '2 5 gms. xff , B 6o'22 gms. / a 57*15 cm. c « 45*55 cm. T - 5*67 X lO*. CHAPTER VII. MECHANICS OP LIQUIDS AND GASES. Measurenunt of Fluid Pressure, The pressure at any point of a fluid is theoretically measured by the force exerted by the fluid upon a unit area including the point The unit area must be so small that the pressure may be regarded as the same at every point of it, or, in other words, we must find the limiting value of the firaction obtained by dividmg the force on an area enclosing the point by the numerical measure of the area, when the latter is made indefinitely small This theoretical method of measuring a pressure is not as a rale carried out in practice. On this system of measure- ment, however, it can be shewn that the pressure at any point of a fluid at rest under the action of gravity is uniform over any horizontal plane, and equal to the weight of a column of the fluid whose section is of unit area, and whose length is equal to the vertical height of the firee surface of the heavy fluid above the point at which the pressure is required. The pressure is therefore numerically equal to the 198 Practical Pf^sics. [Ch. VIL § a^ weight of ph units of mass cf the fluid, where p is the mean density of the fluid, h the height of its free surface above the point at which the pressure is required This pressure expressed in absolute units will be gp^ where g b tlie numerical value of the acceleration of gravity. If the fluid be a liquid, p will be practically constant for all heights ; g is known for different places on the earth's surface. The pressure will therefore be known if the height h be known and the kind of liquid used be specifled. This suggests the method generally employed in practice for measuring fluid pressures. The pressure is balanced by a pressure due to a column of heavy liquid— e.g. mercury, water, or sulphuric acid — and the height of the column necessary is quoted as the pressure, the liquid used being specifled Its density is known from tables when the tem- perature is given, and the theoretical value of the pressure in absolute units can be deduced at once by multiplying the height by g and by p, the density of the liquid at the tempe- rature. If there be a pressure 11 on the free surface of the liquid used, this must be added to the result, and the pressure required is equal to II+^pA Example, — ^The height of the barometer is 755 mm., the temperature being 15^ C.: find the pressure of the atmosphere. The pressure of the atmosphere is equivalent to the weight of a column of mercury 75*5 cm. high and I sq. cm. area, and ^-981 in C.G.S. units. The density of mercury is equal to 13*596 (i-*oooi8x 15) gm. per c.c. In the barometer there is practically no pressure on the free surface of the mercury, hence the pressure of the atmosphere -981 X 13-596 (i -.00018 X 15) X 75-5 dynes per sq. cm. 24. The Mercury Barometer. Barometers are of various forms ; the practical details given here are intended to refer to the Fortin Standard Ch. Vll. § 24.1 Mcclianics of Liquids and Gases, 199 Barometer, in which the actual height of the column of mercury, from the surface of the mercury in the cistern, is measured directly by means of a scale and vernier placed alongside the tube. The scale is only graduated between twenty-seven and thirty-two inches, as the barometric height at any ordinary observatory or laboratory is never outside these limits. To set and read the Barometer. The barometer must first be made to hang freely, by loosening the three screws at the bottom of the frame, in order that the scale may be vertical The mercury in the cistern must be brought to the same level as the zero point of the scale. This zero point is in- dicated by a small ivory point ; and the extremity of this point must first be made to coincide with the surface of the mercury. This is attained by adjusting the bottom of the cistern by means of a screw which projects from the bottom of the barometer; raising this screw raises the mercury surface. On looking at the surface a reflexion of the pointer is seen. Raise the surface until the end of the pointer and its reflected image appear just to touch. Then the mercury surface and the zero of the scale are at the same level The upper surface of the mercury is somewhat convex. In taking a reading, the zero of the vernier must be brought to the same level as the top point of this upper surface. Behind the barometer tube is placed a sheet of white paper, and by raising the vernier this can be seen, through the tube, between it and the upper surface of the mercury. Lower the vernier untiV looking horizontally^ it is just impossible to see the white paper between it and the top of the meniscus ; then the zero of the vernier coincides with the top of the mercury column. To be able to make sure that the eye looks horizontally the vernier is provided with a brass piece on the opposite side of the tube, the lower edge * See Frontispiece, fig. a. 200 Practical Physics, [Ch. VII. § 24. Fxo. is- of which is on the same level as the lower edge of the vernier when the scale is vertical. By keeping the eye always in a line with these two edges we know that the line of sight is horizontal, and thus avoid error of parallax. Of course a glimpse of white may be obtained at the sides, owing to the curvature of the meniscus, as in the figure. The scale is in inches, and is divided to twentieths. Twenty-five divisions of the vernier are equal to twenty- four of the scale ; the instrument therefore reads to 5ooths of an inch. To read it rapidly ; divide the reading of fractions of the inch on the scale by 2 ; the result is in tenths of an inch ; multiply the vernier reading by 2 ; the result is in thousandths of an inch. Thus suppose that the scale reading is 30 inches and three divisions. This is 30*15. The vernier reading is 13, and this is '026 inch ; the reading then is 30*176 inches. If the scale is of brass and is graduated into inches which are * correct 'at 62° F., the corresponding length in miDi- metres on the same brass, * correct ' at o** C, would be given by the annexed table. Thus 30*176 inches = 766*45 mm. YemleT" WM fi j|4 E .SedU ■^30 I in. . , , = 25*392 mm. 6in. . • - 152*344 mm. 2 » . . . - 50785 » 7 „ . . - 177736 n 3 •> • . . - 76*177 n 8 „ • . » 203*128 „ 4 >« . . . - 101-569 „ 9 „ • . - 228*521 „ 5 >» • , . - 126*952 „ 30 „ . . - 761*769 ,, Correction of the Observed Height for Temperature^ &*c. The height thus obtained requires several corrections. (i) Mercury expands with a rise of temperature, and we must therefore reduce our observation to some standard temperature, in order to express the pressure in comparable measure. The temperature chosen is o^ C, and th« co- Ch. VIL § 34.] Mechanics of Liquids and Gases, 201 efficient of expansion of mercury is '000181 per i^ C Thus, if / be the observed height and / the temperatiure, the height of the equivalent column at o^ C is /(i — '000181/). In applying this correction, it iF. very often sufficient to use the mean value, 760 mm. for /, in the small term '000181 //. Now76ox*oooi8is=*i38. Then we can get the corrected height with sufficient approximation by subtracting from the observed height '138 x /. Thus if the observed height be 766*45 mm. and the temperature 15®, the true height, so far as this correction only is concerned^ b 766*45 — 15 x*i38=766*4S—2-o7=764'38 mm. (2) The same rise of temperature has caused the brass scale to expand, so that the apparent height of the columii is on that account too short To obtain the true height we must add to the observed height /, the quantity /^/, p being the coefficient of linear expansion of brass.^ Now j3 =s '000019. The complete correction then due to both causes will be —('000181— '000019) //, and the true height is /— ('00018 1— '0000 19) //or /— (•000162)//. If in the small term, ('000162)//, we take the mean value, 760 mm., for /, the true height is ^, whereas/— '123/. Thus in our case (/=i5**), ^^^66^^$ — i-85=764-6o mm. (3) Owing to the capillary action between the glass of the tube and the mercury, the level of the mercury is depressed by a quantity which is roughly inversely proportional to the diameter of the tube. The depression is not practically of an appreciable amount unless the tube has a diameter less than a centimetre. In the instrument in the Cavendish La- boratory the tube is 5*58 mm. in radius, and in consequence the top of the meniscus is depressed by about *o2 mm. ; we must therefore add this to the observed height, and we find that the corrected value of the height is 764*62 mm. (4) Again, there is vapour of mercury in the tube, which > The correcti<m is made to 6° C because millimetre graduatioii if generally made to be ' correct ' at that temperature. If the scale correc- tioo is applied to the indies it must be computed from 6a^ F. 202 Practical Physics, [Oh. VIL § 24. produces a pressure on the uppei suilace of the column. It is found that at temperature / this may be practically taken to be equivalent to 'ooa x / mm. of mercury. Thus, if the temperature be I5^ we must on this account add to the observed height '03 mm., and we obtain as our corrected height 764*65 mm. This is the true height of the column of mercury at standard temperature, which gives a pressure equivalent to the pressure of the atmosphere at the place and time in question. (5) Now the weight of this colimin is balanced against the pressure of the air. The weight of the column will depend on its position relatively to the earth. We must therefore determine the height of the column which at some standard position will weigh as much as our column. We take for that standard position sea-level in latitude 45®. Let ^0 ^ the value of the acceleration due to gravity at this position, ^0 the height of a column weighing the same as our column b ; g the acceleration due to gravity at the point of observation. Then, since the weights of these two columns are the same, we have b^g^^—bg^ and therefore bo=bg/go. Now it is known from the theory of the figure of the earth that if ^ is the height above the sea-level in metres and ^ the latitude of the place of observation, Hence ^= I — '0026 cos 2^— •0000002^. ^0 ^Q = ^(l — '0026 cos 2<^ — '0000002^). Experiment — Read the height of the standard barometer, and correct to sea-level at 45^ lat 35. The Aneroid Barometer. In the aneroid barometer at the Cavendish Laboratory each inch of the scale is divided into fiftieths, and there is a vernier,' twenty half-divisions of which equate with * See Frontispiece, 6g. 4. Ch. VII. § 25.] Mechanics of Liquids and Gases. 203 twenty-one of the scale ; the vernier reads, therefore, by estimation to thousandths of an inch. On the vernier each division must be counted as two, only the even divisions being marked. The aneroid is set by comparison with a corrected mer- cury barometer, to give the true pressure at the time of the observation. If properly compensated for temperature, it would continue to give the true barometric height at any other station, even if the temperature changes. To read the aneroid, set the zero of the vernier exactly opposite the end of the pointer, and read the inches and fiftieths on the scale up to the vernier zera Multiply the fractional divisions by 2 ; the result is in hundredths of an inch. Read the vernier, and again multiply by 2 ; the result is in thousandths of an inch. (The numbers marked on the scale give tenths of an inch; those on the vernier thousandths.) Thus the scale reading is between 30 and 31, the pointer standing between divisions 12 and 13. The scale reading, therefore, is 30*24. When the zero of the vernier is opposite the pointer there is a coincidence at division 8 of the vernier; the vernier reading is, therefore, •016, and the exact height is 30*256. To measure the height between two stations with the aneroid) take the reading at the two stations and subtract The difference gives the pressure in inches of mercury of the column of air between the two. Thus suppose that at a lower station the reading of the aneroid is 30*276, and the difference in pressure is that due to 0*020 inch of mercury ; this is equivalent to 0*51 mm. The specific gravity of mercury is 13*60 ; thus "51 mm. mercury is equivalent to 'Six 13*60 mm. of water at 4® C To find the true height of the column of air which is equivalent in pressure to this, we must divide by the specific gravity of air at the temperature and pressure of obser- vation. This may be determined when the pressure and 204 Practical Physics. [Ch. VIL § as. temperature have been observed, by calculation from the data^ven in No. 36 of Lupton's * Tables.' If the difference of height is not great the pressure oi the air between the two stations may, for this purpose, be taken to be the mean of the aneroid readings at the two stations, properly corrected by reference to the mercury standard. For the temperature, if there is any considerable difference between the thermometer readings at the two stations, some judgment must be used in order to get a mean result which shall fairly represent the average temperature of the air between the twa When these observations have been made, we are in a position to calculate the specific gravity of dry air under the given conditions. Since tho atmosphere always contains more or less moisture, a correc- tion must be applied. Since the specific gravity of aqueous vapour referred to air at the same temperature and pressure is I, the correction may be made by calculating what would be the specific gravity of the dry air if its pressure were diminished by an amount equivalent to three-eighths of the pressure of the water vapour it contains, as determined by observation of the dew-point or other hygrometric method. This correction is often so small as to produce no appre- ciable effect within the limits of accuracy of the pressure readings. Thus if the mean of the pressure observations be 768 mm., and the estimated mean temperature 15^ C, the specific gravity of dry air would be 0*001239, and if the ot^erved pressure of aqueous vapour be 10 nun., the corrected specific gravity would be 2 — "^^ '^ X -001239, or -001233. Hence the height of the column of air between the two stations is -^ ^ — mm., or 563 cm. •001233 Ch, VII, 5*6-1 MecJianics of Liquids and Gases. 203 For a method of extending the application of barometnc obserradons to the measurement of comparatively greater heights we may refer the reader to Maxwell's ' Heat,' chap. Tiv. Experiment.— "Rtzi the aneroid and determine from yonr observation of the standard the correction to be applied to the aneroid to give the true reading. Measure the height of the laboratory from the basement to the tower. 36. The Toliimenom«t«r. The apparatus (fig. iti) consists of two glass tubes placed invertical positions ^^^ _^ against a scale. The one tube (c d) is fixed, and has at the top an elbow with a "^ screw by means of which . a small flask b can be ' fastened on. [In another | form of the apparatus the ' tube D c ends in a bulb f, which opens into a funnel- shaped space. The upper edges of the funnel are ground fiat, and the irfiole can be closed so as to be air-tight by means of a ground glass plate and grease.] The other glass tube is attached to a sUding piece movable along the vertical scale ; the lower ends of the two tubes are connected by means of a piece of flexible india- nibber tubing ; this and portions of the glass tubes contain 2o6 Practical Physics. [Ch. Vll. § 26. mercury, which so long as the end £ is open stands at the same level in the two tubes. The instrument is supported on three levelling- screws, by means of which the scale can be set vertical. The whole apparatus should stand in a wooden tray, which serves to catch any mercury which may unavoidably be spilt The following experiments may be made with it : — (i) To test Boyl^s Law, viz,, ifv be the Volume andp thi Pressure of a Mass of Gas at constant Temperature, then s p is a constant. We shall require to know the area of the cross section of the tube cd. For this purpose suppose the flexible connection between the bottoms of the tubes is removed, and replaced by a short piece of tubing closed with a pinch- cocL Fill this tubing and the glass tube above it with mercury up to some convenient division of the scale, taking care that all the air-bubbles are removed ; this can generally be done by tilting the apparatus or by means of an iron wire. The mercury should be clean and dry, otherwise it will stick to the glass. Now open the pinch-cock and allow some of the mercury to escape into a weighed beaker. When a convenient quantity has run out close the pinch- cock, and again read the level of the mercury on the scale ; let the difference of the two levels be / centimetres, and let the area of the tube be a square centimetres. The volume of mercury which has run out, is la cubic centimetres, and if p is the density of mercury in grammes per cc, its mass is f>/cf grammes. Weigh the mercury in the beaker ; let its mass be m grammes; then p/a ssM, .*. a =. w /p /. The density of mercury is very approximately 13*59 grammes per cc, and hence if we measure m and / we find Ch. VII. § 26.] Mechanics of Liquids and Gases, - 207 a, the area of the cross section. The above assumes the area to be constant throughout the length of the tube ; if this condition is not sufficiently nearly satisfied the tube must be calibrated (see § 8). When the value of a is known the connexion between the tubes at b and d may again be made and the apparatus filled with mercury, which can be poured into the open end of one of the tubes though a funnel ; while this is being done the flask e should be removed, the end of the tube being left open, and the mercury should be poured in until it reaches nearly to the top of the tube d c. Now screw on the flask or close the end of the tube with the glass plate. If this is done carefully the merciu7 will stand at the same level in the two tubes, and the air in the bulb will be at the same pressure as the air outside. Let the volume of the air be v cubic centimetres — we shall shew how to find v shortly— and let the height of the mercury barometer which measures the pressure be h centi- metres. Read on the scale the level of the mercury in the tube D a Let it be a centimetres. This is facilitated by having a vertical piece of looking-glass at the back of the tube; by placing your eye so that the mercury and its image appear at the same level errors of parallax are avoided. It is convenient to have the tube mounted so that a piece of looking-glass can be inserted between it and the frame and held pressed against the vertical stand in the proper position while making an observation ; in some instruments the scales are engraved on looking-glass. Now lower the sliding tube a b. The mercury falls in both tubes, but to a less amount in the tube d c than in the other. Read on the scale the level in each tube. Let that in d c be 3 cm., and in a b, 3^ cm. Then the volume of the enclosed air has increased from V to y -f a (a — 3), while, since the difference in levels in 208 Practicul Physics. [Ch. vii §26. Fig. z6«. the two tubes \&b ^V cm., the pressure of the enclosed air is now measured by ^ — (3 — b'). Again alter the position of the sliding tube and make a similar set of ob- servations. Now write down from a table, or by actual calculation, the reciprocals of the pressures, and then plot a curve, taking as abscissae lines proportional to the additional volume rt (a — ^) of the tube occupied by the air, and for ordinates the reciprocals of the pressiures or numbers pro- portional to I / {^ — (^ — ^)} • If the measurements be made with care, it will be found that the curve obtained by joining the points thus found is a straight line such as PBA in fig. 16a, cutting the vertical axis in b and the horizontal axis in a, a point on the nega- tive side of the ori- gin. Now o B is the reciprocal of the ori- gnud pressure, the additional volume is zero, the actual volume is v, and the barometric height is given by i/o e. At A the reciprocal of the pressure is zero ; the pressure is therefore infinitely large, and the actual volume infinitely small. Thus the distance o A is — v, and the distance a n measures the actual volume v + « (« — ^), when the reciprocal of the pressure is p n. But since the curve given by the experiments is a straight line, a n is proportional to ? n, or the volume is inversely proportional to the pressure ; in other words, the product of the pressure and volume is constant Before taking any readings to determine the difference of pressure it is well to wait a few minutes and notice if the Ch. VII, § 26.] Mechanics of Liquids and Gases. 209 le\'els remain the same. If they do, we may feel sure there ts no leak at the joints. (2) To determine by means of the Volumenometer the Density of a Solid, The method is useful in the case of solids soluble in or affected by water. The solid should be broken into frag- ments sufficiently small to go into the flask e. Determine the volume of the flask and a small portion of the tube d e down to some convenient mark, as above. We can do this from one pair of observations if we assume Boyle's law to be true, for then we have V ^= { v + a(tf-^)} {>i-(^-.^)}. ■ Now weigh the solid, and place it in the flask ; deter- mine as before the volume of the portion of the flask not occupied by the solid, together with that of the tube d e down to the same mark ; let this volume be v'. Then v— v' is the volume of the solid in cubic centi- metres. But the mass of the solid has been found in grammes ; dividing this by the volume, we have the density in grammes per cubic centimetre. If the second form of apparatus be used with the bulb and funnel, it is best to make two marks on the tube, one at f, between the bulb and funnel, the other at g, just below the bulb, and to determine the volume between these marks in the same way as the volume of part of the tube was found. Let this volume be v cc Then in using the instrument to measure the volume of a solid it is filled with mercury up to the upper mark f at the atmospheric pressure, and then, the funnel being closed, the sliding tube is lowered until the mercury falls to the lower mark. Thus the volume of the contained air 2IO Practical Physics, £Ch. VII. § 261 increases by 9, which takes the place of a{a —b) In the formula, and we have The method will give accurate results only in the case in firhich the volume of the solid is considerable ; it should nearly fill the flask. ExptfitmtUs, (i) Test Boyle's law, and measure the volume of the small flask attached to the volumenometer. (2) Determine the density of the given glass beads. Enter results thus : — Area of cross section of tube 1*01 sq. cm. Four observations of increase of volume and corresponding pressures, made and plotted on curve shewn. Volume deduced from diagram . . • 155 c.c Volume by calculation from one observation : — Division to which tube is filled, a . . 90 cm. Division to which mercury falls, b . • 72 » Level of mercury in sliding tube, b' . • 64 ,, Height of barometer, A . . • • 76 ,9 a^b • • • 18 cm. b—b' • • • 8 n Volume •>...•• 154*5 cc H. Capillarity. To Measure the Surface Tension of a Liquid by the height it rises in a Capillary Tube, If a narrow tube is dipped into a hquid which wets it, the liquid rises in the tube and stands at a higher level than in the containing vessel. From this we infer that the particles of the liquid in the neighbourhood of the surfrure are in a diflferent condition from those in the interior of its Ch. VIL S &] Mechanics of Liquids and Gases. 2i i mass, and, in consequence, possess a greater amount of potential energy (see MaxweU's 'Theory of Heat,' chap, xx). The effect may be represented by supposing that the sur£Eu:e fihn of any liquid is under tension, so that if we draw any line across it we may conceive the portion of the film on one side of the line to act on the portion on the other side with a definite force. The amount of this force per unit of length is found to be a constant for the surface of separation of any two given fluids, and it may be shewn to be equal to the amount of surface energy per unit of area which the fluids possess. If now we have three fluids meeting at a point, there will at that point be three definite forces — the tensions of the three surfaces of separation, and in order that there may be equilibrium the surfaces must meet at definite angles. Now let one of the substances r be a solid, and let a and b be the other two. Let t.^ (fig. xvii) represent the tension between the surl^ces of a and ^, and let this surface at o make an angle a with the surface of c. Then, resolving the forces at o parallel to the surface, we have for equilibrium Ta» cos a = Tfte — Tc«. This equation determines a, the angle of capillarity. If Tftc — Tea ^s greater than t.j, no such angle as a can be found ; the liquid is said to wet the surface of the solid, and wHl run all over it unless prevented by other forces, such as gravity. The system of two fluids and the solid tends to set itself, so that its whole eneigy is as small as possible. And since the surface energy of the water-air surface is less than that of the air-glass surface in the case of water in contact with glass, the water tends to cover the glass. If the glass sur&ce be vertical the water as it creeps P 2 2I» Practical Phystcs. [Cb. VIL J m up the surface f^ins potential energy, and equilibrium is reached when the gain of potential energy due to the rise of water is equal to the loss due to the diminution of air-glass surface. To determine the surface tension of a liquid we require to know the density of the liquid, the diameter of the tube, the angle of contact, and the height the liquid rises. Let the section of the tube be a circle of radius r. The circumference of this is jir r, and at each point of this circumference there is a force t per unit of length acting at an angle a with the vertical. The total vertical force is z B- r . T cos a. If A be the height of the volume of liquid raised, measured from the flat surface of the liquid in the vessel to the bottom of the meniscus in the tube, and tlie weight of the very small portion forming the men- iscus be neglected, then the weight of liquid raised is .'. XTT r cos a=-rpgr^A, •'• T = ipf rA sec a dynes per cm. In practice the method is only used with a liquid, such as water, which wets the glass, and then a ss o, sec a = I, .•. "T^^pgrA dynes per cm. To perform the experi- ment a finely divided scale (a b, fig. xviii) must be placed in a vertical position, with one end dipping into the beaker c, which is to contain the liquid ; the scale may most conveniently be of glass divided into millimetres and Ch. VII. § H.] MecJtanics of Liquids and Gases, ? t 3 some 30 cm. long. It may be adjusted to a vertical position by means of a plumb-line d. The capillary tube is attached to the scale by two elastic bands; the scale should be illuminated from behind with a good light, which may be thrown on to it by a minoi if requisite. The capillary tube is prepared by softening a piece of clean glass tubing in a blow-pipe flame, and drawing it out until the diameter is comparable with about half a milli- metre. The ends of the tube should be sealed until it is wanted for use. When the scale and light have been arranged, fasten the tube to the scale in a vertical position so that it may dip into the water, and open the two ends ; the water wiU rise in the tube. When the rise has ceased, dip the tube slightly further into the water and then raise it a little. This will ensure that the tube is wetted above the level of the water it contains. Now read on the scale the height to which the water has risen ; read also the position of the horizontal water- sur£u:e in the beaker. If there is any difficulty in doing this directly, it may be overcome by fastening a fine needle in a suitable clip and lowering it gently near the scale until it just touches the water ; the level of the needle-point can then be found. The difference in these two readings gives the height h. The height so foimd can, if required, be afterwards corrected for the meniscus by adding one-third of the radius of the tube. We have next to measure the diameter of the tube ; for this purpose it must be carefully cut in two close to the point to which the water rose. This may be done by holding the tube against the finger and 'gently drawing a fine file with a sharp edge across it The tube is then mounted with a little wax on a suitable stand or clip so that the section is in the field of a good microscope with a micrometer scale in the eyepiece. The value of the 214 Practical Physics. [Ch. vil. § h, divisions of the eye-piece micrometer must have previously been determined by viewing through the microscope a finely divided scale, and counting the number of divisions of the eye-piece micrometer which coincide with one division of the scale. For this purpose a scale on glass, divided to half or quarter millimetres, is useful, or an ordinary stage micro- meter having loo divisions to the inch may be used. If, then, we find that a certain number of divisions — say 52 — of the eye-piece micrometer coincide with 2 divisions of the scale, then i division of the micrometer is equivalent to 1/26 X 100 of I inch. When the section of the tube is viewed through the microscope it will probably be seen that it is not circular ; if it appears distinctly oval the results of the experiment will not be very satis&ctory ; otherwise by observing the diameter in several directions— say four — inclined at angles of about 45** to each other, and taking the mean, we shall obtain a result not far from the truth. It will usually happen that the diameter of the tube is not an exact whole number of divisions of the scale, but the divisions can be subdivided by eye to quarters, or even to tenths, and in this way a fairly accurate value for the diameter of a small tube may be found. The diameter may also be measured, and this more easily, by the aid of a good travelling microscope. Experiment — Determine by means of a capillary tube the surface tension of water. Enter results thus : — Values of ^, 7'39 - 1 •! i - 6*28 cm. 7*40 -I'll -6-29 „ 2^1 ■■ 5*^25 - 71 - 4*415 divs. of micrometer. 2r3 = 3-8o + x>7-3'87 lvalue of a division of micrometer scale » x>i23 cm. T" 78*3 dynes per cm. CH. VIL $ 1.] Mecfianics of Liquids and Gases. 315 I. Worthiiigton*8 Capillary Mnltiplier. If a substance which is wetted by a liquid is dipped into the h'quidy the liquid is raised by the action of the surface tension, and there is a downward force on the substance equal to the weight of liquid raised. Thus, in the last experiment the force raising the column of liquid is 2ir rT, and in opposition to this there must be a force acting in the downward direction on the tube. The apparent weight of the tube is increased by this amount If this increment of weight be determined, and the radius of the tube be known, we have another method of finding x. In the case of a narrow tube the apparent increase of weight would be very small ; but suppose a flat strip of some sub- stance — say a thin sheet of glass or of platinum foil — be immersed vertically in the fluid so that the lower edge of the strip is horizontal and level with the undisturbed sur- £u:e, and let n^ be the weight of liquid raised,/ the length of the strip with which it is in contact, then the surface tension is wjp. Now by making the strip sufficiently long p may be made considerable, and the apparent increase in weight large enough to be found with accuracy. It is necessary that the lower edge of the strip should be level with the undisturbed surface of the liquid, for if the strip dips beneath this there is a correction necessary on account of the buoyancy of the liquid, while if the base of the strip IS raised above the free surface there is a traction to correct for, due to the adhesion of the liquid to the horizontal edge of the strip. The method in some form or other is an old one. Re- cently Pro£ A. M. Worthington has given an account of various improvements in carrying it out, which we proceed to describe in detail A strip of thin platinum foil, about 50 cm. long and some 6 or 8 cm. wide, is rolled into a spiral coiL The 2l6 Practical Physics. fCH. VII. § f. successive convolutions of the coil are separated from each other by a number of glass beads, about 2 mm. in diameter, strung on platinum wire (see fig. xix). The beads are made of combustion tubing, which is first cut into lengths of about 20 cm. A number of such lengths Fig. xix. of the most Uniform thickness are selected, and these are cut into pieces 2 cm. long. Of the beads thus formed a strip 50 cm. in length is put together by passing through each in turn in opposite directions the two ends of a fine platinum wire. The strip of beads Is then laid on the foil, the length of the strip being parallel to that of the foil, and the whole is rolled into a spiral and secured with platinum wire ; the convolutions of this spiral are thus 2 mm. — the diameter of the beads — apart, while the length of the strip which is in contact with the liquid is, taking both surfaces of the foil into account, 100 cm. The beads should be at least 3 cm. above the lower edge of the foil in order that the liquid which rises between the convo- lutions may not reach so high as to wet the glass. The whole is made of platinum and hard glass, in order that it may readily be cleaned by beating to a bright red in a Bunsen flame. The foil should not be more than '0025 cm. thick. The coil is then suspended from one arm of a balance, and the suspending strings are adjusted so that its under surface may be horizontal (see fig. xx). In order to prevent its being drawn too far below the surface of the liquid when in use, stops, as shewn in the figure, are fitted to the balance. These consist merely of two stout pins stuck in a cork and held in place by a clip. The coil is then carefully counterpoised, and a wooden block c is placed under the pan d in such a position that when the pan is held down on the block the beam may be Ch. VIL § L] Mechanics of Liquids and Gases. 217 horizontaL Weights are then placed in d to hold it in this position. The liquid is then placed in a beaker on a small table fitted with a vertical screw motion under the coil (Mr. Worthington made use of the end of an optical lantern with a card on the top), and the screw is turned, thus raising Fig. XX. the liquid until it wets the foil. Some of the liquid is drawn up between the convolutions, and the screw is ad- justed until the under-surface of the coil when the beam is horizontal is level with the undisturbed surface of the water. This adjustment can be made with considerable accuracy, and Mr. Worthington has shewn that an error of i mm. will not cause an error of more than ^^J^ in the value of the sur^ce tension of alcohol. On removing the weights in d the coil is drawn as far below the liquid (some 4 or 5 mm.) as the stops will allow. c is then removed, and weights put into d to restore equili- brium. The difference between these weights and the weights onginaUy used to counterbalance the coil gives the total downward pull due to the surface tension. On dividing this by the total length of the line of separation between the liquid and the foil — 100 cm. if the strip of platinum be 2i8 Practical Physics. [Ch. vil § l 50 cm. long — we get the value of the surfisice tension. The adjustment for level should be made more than once, and the observations of weight repeated. Experiment — Determine the surface tension of water by the capillary multiplier. Enter the results thus :— Length of strip in contact, given with instrument, 100 cm. Weight to counterpoise strip . . . 37*258 gm. r44-9i8 „ Weights with coil in water • . . •< 44'9M » L44-90I n Mean * • 44*911. Total downward force . . 7*653 „ Surface tension ^ ^^3 ^ 9 — dynes per cm. - 75 dynes per cm CHAPTER VIII. ACOUSTICS. Definitions^ &*c. A MUSICAL note is the result of successive similar dis- turbances in the air, provided that they follow each other at regular intervals with sufficient rapidity. Similar dis- turbances following each other at regular equal intervals are said to be periodia The interval of time between successive impulses of a periodic disturbance determines the pitch of the note produced — that is, its position in the musical scale. The pitch of a note is therefore generally expressed by the number of periodic disturbances per second required to pro- duce it This number is called the ' vibration number/ or ' frequency ' of the note. It generally happens that any apparatus for producing a note of given frequency produces at the same time notes of other frequencies. The result is a complex sound, equivalent to the combination of a series of simple sounds or tones. Ch, VHL § 27.] Acoustics. 219 The simple tones of which the complex sound may be re- garded as consisting are called ' partial tones ; ' the gravest of these — that is, the one of lowest pitch — is called the 'funda- mental tone ' of the sounding body, and the others are called ' upper partials.' A note which has no upper partials is called a pure tone. By means of suitable resonators the different partial tones of a complex note may be made very clearly audible. For many musical instruments, as organ-pipes, string instruments, &c., the ratio of the vibration frequency of any upper partial tone to that of the fundamental tone is a simple integer, and the upper partials are then called 'har- monics ; ' for others, again, as for bells, tuning-forks, &c, the ratios are not integral, and the upper partials are said to be mharmonic 27. To eompare the Frequeneies of two Tuning-forks of nearly Identical Pitch, and to tune two Forks to unison. A tuning-fork mounted upon a resonator — ^a wooden box of suitable size — furnishes a very convenient means of obtaining a pure tone ; the upper partials, which are gene- rally heard when the fork is first sounded, are not reinforced by the soimding box, and rapidly become inaudible, while the fundamental tone is, comparatively speaking, permanent When two forks which differ only slightly in pitch are set in vibration together, the effect upon the ear is an alternation of loud sound with comparative silence. These alternations are known as beats, and they frequently are sufficiently well marked and sufficiently slow for the interval of time between successive beats to be determined with considerable accu- racy by counting the number occurring in a measmed interval of time. It is shewn in text-books on sound ^ that the number of beats in any interval can be inferred from the vibration num- ■ Deschanel, Natural Philosophy^ p. 813 ; Poynting & Thomson, Sound I TyDdall» On Sounds p. 261. 220 Practical Physics. [Cm. vin. § 27. bers of the two notes sounded together, and that if n be the number of beats per second, «, »' the frequencies of the two notes, n being the greater, then We have, therefore, only to determine the number of beats per second in order to find the difference between the frequencies of the two notes. This may be an easy or a difficult matter according to the rapidity of the beats. If they are very slow, probably only few will occur during the time the forks are sounding, and the observer is liable to confuse the gradual subsidence of the sound ¥rith the duninution of intensity due to the beats. If, on the other hand, there are more Uian four beats per second, it becomes difficult to count them without considerable practice. The difficulty is of a kind similar to that discussed in § 1 1, and we may refer to that section for further details of the method of counting. In order to determine which of the two forks is the higher in pitch, count the beats between them, and then lower the pitch of one of them by loading its prongs with small masses of sheet lead, or of wax (softened by turpen- tine), and observe the number of beats again. If the number of beats per second b now less than before, the loaded fork was originally the higher of the two ; if the number of beats has been increased by the loading, it is probable that the loaded fork was originally the lower ; but it is possible that the load has reduced the frequency of the higher fork to such an extent that it is now less than that of the unloaded second fork by a greater number than that of the second was originally less than that of the first It is safer, therefore, always to adjust the load so that its effect is to diminish the number of beats per second, that is, to bring the two forks nearer to unison ; to d«r so it must have been placed on the fork which was originally of the higher pitch. In order to adjust two forks to unison, we may lower the CH. VIII. ^ 27.] Acoustics. 221 pitch of the higher fork by weighting Its prongs until the beats disappear ; the difficulty, already mentioned, when very slow beats are observed occurs, however, in this case, and it is preferable to use a third auxiliary fork, and adjust its pitch until it makes, say, four beats a second with that one of the two forks which is to be regarded as the standard, noting whether it is above or below the standard. The second fork may then be loaded so that it also makes four beats a second with the auxiliary fork, taking care th^t it is made higher than the auxiliary fork if the standard fork is sa The second fork will then be accurately in unison with the standard — ^a state of things which will probably be shewn by the one, when sounded, setting the other in strong vibration, in consequence of the sympathetic reso- nance. A tuning-fork may be permanently lowered in pitch by filing away the prongs near their bases ; on the other hand, diminishing their weight by filing them away at their points raises the pitch. Such operations should, however, not be undertaken without consulting those who axe responsible fcnr the safe custody of the forks. Experiment. — Compare the firequencies of the two given forks A and B by counting the beats between them. Determine which is the higher and load it until the two are in unison. Enter results thus : — Number of beats in 25 sees. , . 67 Number per sec .... . . 27 „ „ (A loaded) 3-3 „ n (B loaded). • . • 2*1 B is the higher fork. Number of beats per sec between a and the auxiliary fork c 3*6 Number of beats per sec. between B (when loaded) and the auxiliary fork c » • 3*6 222 Practical Physics, [Ch. VIIL f la 28. Determination of the Vibration Frequency of a Vote by the Siren. A siren is essentially an instrument for producing a musical note by a rapid succession of puffs of air. The simplest form of siren is a large circular cardboard disc, provided with perforations arranged in circles concentric with the disc. The puffs of sur may be produced by blow- ing through a fine nozzle on to the drde of holes while the disc is maintained in rapid rotation. In order that the dis- turbances produced by the puffs of air passing through the holes may be periodic (see p. 218), the holes must be punched at equal distances from each other, and the disc must be driven at a uniform rate. If the pressure of the water-supply of the laboratory is sufficiently high, a small water-motor is a convenient engine for driving the disc, which must be mounted on an axle with a driving pulley. If the diameter of the disc is considerable, so that a large number of holes can be arranged in the drde, a rotation of the disc giving four revolutions per second is quite sufficient to produce a note of easily recognisable pitch. The revolutions in a given interval, say, one minute, can be counted, if a pointer be attached to the rim of the disc, and arranged so that it touches a tongue of paper fixed to the table once in every revolution. The number of taps on this paper in a given time is the number of revolu- tions of the disc Suppose the number of taps in one minute is n, and the number of holes in the circle which is being blown is n, then the number of puffs of air produced per minute is n if, and hence the number per second is Nif/6a The disc is genendly provided with a series of concen- tric rings of holes differing in the number of perforations in the drde, so that a variety of notes can be blown for the same rate of rotation of the disc. In the more elaborate forms of the instrument a metal Cb. vilL § 28.] Acoustics. 223 disCy which is perforated with holes arranged in concentric cirdesi is mounted on a spindle so that it can revolve parallel and very near to the lid of a metallic box, which can be supplied by air from foot-bellows. The lid of this box is perforated in a manner corresponding to the revolving discy but the holes in either opposing plate, instead of being bored perpendicularly through the metal, are made to run obliquely, so that those in the upper disc are inclined to those in the lower. When air is driven through the box it escapes through the holes, and in so doing drives the disc round. The disc may thus be maintained in a state of rotation, and if the pressure of the air be maintained con- stant the rotation will be uniform. In driving the siren a pressure-gauge, consisting of a U-tube containing water should be in connection with the tube conveying the air from the bellows to the instrument ; the blowing should be so managed as to keep the pressure of wind as indicated by this gauge constant The number of revolutions of the spindle carrying the revolving disc is generally indicated on two dials— one showing revolutions up to a hundred, and the other the number of hundreds — by a special counting arrangement This arrangement can be thrown in and out of gear at plea- sure, by pushing in one direction or the opposite the knobs which will be foimd either in front or at the sides of the box which carries the dials. The process of counting the revolution of the spindle is then as follows : — First read the dials, and while the rota- tion is being maintained constant by keeping the pressure constant, as indicated by the gauge, throw the counting apparatus into gear as the second hand of a watch passes the zero point; throw it out of gear after a minute has been completed, and read the dials again. The difference of readings gives the number ot revolutions of the spindle in one minute ; dividing by 60 the number per second is obtained. 224 Practical Physics. [Ch, VIIL § 28. To obtain the number of pufTs of air we have to multiply by the number of holes in the revolving circle. In the modification of the siren by Dove there is a series of circles of holes, which can be opened or shut by respectively pushing in or pulling out plugs in the side of the box. The number of holes in the circles opened or shut by the re- spective plugs is stamped on the head of the plugs them- selves. In Helmholtz's double siroi ^ we have practically two siren discs working on the same spindle ; the box of one of the sirens is fixed, while that of the other is capable of com- paratively slow rotation. By shutting off all the holes of the one box this siren can be used exactly as a single one. We are thus furnished with a means of producing a note of any pitch, within certain limits, and of counting at the same time the number of puffs of air which are required to produce it The note produced by a siren is not by any means a pure tone ; the upper partials are sometimes quite as loud as the fimdamental tone. To measure the vibration frequency of a note by means of the siren, the pressure of air from the bellows must be adjusted so that the siren is maintained at a constant rate of rotation, and giving out a note whose fundamental tone is in unison with that of the given note, one circle of holes alone being open. The condition of unison between the two notes may be attained by starting with the siren considerably below the necessary speed, and, sounding the note at same time, gradually increase the speed of the siren until beats are distinctly heard between the given note and the sirea As the speed of the siren is still further urged the beats become less rapid until they disappear ; the blower should then keep the pressure so constant that the note of the siren remains in exact unison with the given note, and while this constancy b maintained a second observer should measure > For a more detailed description of this instrument, see Tyndall'i Sounds Lecture IL Ch. VIII. § 28.] Acoustics, 225 the rate of rotation of the spindle. The beats which will be heard if the note of the siren is too high or too low serve to aid the blower in controlling the note of the siren. Suppose that the number of revolutions per minute is n, and the number of holes in the open circle /r, then the vibration frequency of the note is n «/6o. The method of procedure with the simpler siren pre- viously described is similar. The speed of rotation depends in that case, however, on the rate of driving of the engine ; the experiment is therefore somewhat simpler, although the range of notes obtainable is rather more limited. The speed can be controlled and kept steady by subjecting the driving string to more or less friction by the hand covered with a leather glove. Care should be taken not to mistake the beats between the given note and the first upper partial of the note of the siren, which are frequently very distinct, for the beats between the fundamental tones. The result of a mistake of that kind is to get the vibra- tion frequency of the note only half its true value, since the first upper partial of the siren is the octave of the funda- mental tone. It requires a certain amount of musical per- ception to be able to distinguish between a note and its octave, but if the observer has any doubt about the matter he should drive the note of the siren an octave higher, and notice whether or not beats are again produced, and whether the two notes thus sounded appear more nearly identical than before. The most convenient note to use for the purpose of thfa experiment is that given out by an organ-pipe belonging to the octave between the bass and middle c's. In quality it is not unlike- the note of the siren, and it can be sounded for any required length of time. For a beginner a tuning-fork is much more difficult, as it is very different in quality from the siren note^ and only continues to sound for a com* paratively short time. Q 226 Practical Physics. [Ch. viii. § aft. If a beginner wishes to find the vibration frequency of a fork by the siren, he should first select an organ-pipe of the same pitch. This can be tested by noticing the resonance produced when the sounding fork is held over the em- bouchure of the pipe. Then determine the pitch of the note of the organ-pipe by means of the siren, and so deduce that of the fork. Experiment. — Find the vibration frequency of the note of the given organ-pipe. Enter results thus : — Organ-pipe — Ut. 2 (i) By the Helmholtz siren: Pressure in gauge of bellows, 5 inches. Revolutions of spindle of siren per minute, 64S. Number of holes open, 12. Frequency of note, 129. (3) By Ladd's siren : Speed of rotation of disc, 3*6 turns per sec. Number of holes, 36. Frequency of note, 13a 29. Determination of the Velocity of Sound in Air by Measurement of the Leng^ of a Eesonance Tabe corresponding to a Fork of known Fitch. If a vibrating tuning-fork be held immediately over the opening of a tube which is open at one end and closed at the other, and of suitable length, the column of air in the tube will vibrate in unison with the fork, and thus act as a resonator and reinforce its vibrations. The proper length of the tube may be determined experimentally. If we regard the motion of the air in the tube as a succession of plane wave pulses sent firom the fork and reflected at the closed end, we see that the condition for resonance is that the refiected pulse must reach the fork Ch. VIII. § 29.] Acoustics, 227 again at a moment when the direction of its motion is the opposite of what it was when the pulse started. This will always be the case, and the resonance will in consequence be most powerful, if the time the pulse takes to travel to the end of the tube and back to the fork is exactly half the periodic time of the fork. Now the pulse travels along the tube with the constant velocity of sound in air ; the length of the tube must be, therefore, such that sound would travel twice that distance in a time equal to one half of the periodic time of the fork. If « be the vibration frequency of the fork, i/« is the time of a period, and if / be the required length of the resonance tube and v the velocity of sound, then 3/ I V 2n or v=s4/n (i) In words, the velocity of sound is equal to four times the product of the vibration frequency of a fork and the length of the resonance column corresponding to the fork. This formula (i) is approximately but not strictly accu- rate. A correction is necessary for the open end of the pipe ; this correction has been calculated theoretically, and shewn to be nearly equivalent to increasing the observed length of the resonance column by an amount equal to one half of its radius. ' Introducing this correction, formula (i) becomes z;=4(/+r/.>, (2) where r is the radius of the resonance tube. This furnishes a practical method of determining v. It remains to describe how the length of the resonance tube may be adjusted and measured. The necessary capability of adjustment is best secured by two glass tubes as A, B, in fig. 17, fixed, with two paper millimetre scales * See Lord Ra) leigh's Sounds vol. ii. § 307 and Appendix A. 228 Practical Posies. [Cii. vm. ) 39, behind them, to two boards arranged to slide vertically up and down in a wooden frajne ; the tubes are drawn out at Fio, 17. the bottom and connected by india- rubber tubing. The bottoms of the tubes and the india-rubber connec- tion contain water, so that the length of the column available for reso- nance is determined by adjusting the height of the water. This is done by sliding the tubes up or down. The position to be selected is the position of maximum resonance, that is, when the note of the fork is most strongly reinforced. The length of the column can then be read off on the paper scales. The mean of a large number of observations must be taken, for it will be noticed, on making the experiment, that as the length of the tube is continuously increased the resonance increases gradually to its maximum, and then gradually dies away. The exact position of maxi- mum resonance is therefore rather difficult of determination, and can be best arrived at from a number of observations, some on either side of the true position. From the explanation of the cause of the resonance of a tube which was given at the outset, it is easily seen that the note will be similarly reinforced if the fork has executed a complete vibration and a half, or in fact any odd number of half-vibrations instead of only one half- vibration. Thus, if the limits of adjusCment of the level of the water in the tube be wide enough, a series of positions of maximum resonance may be found. The relation between the velocity of sound, the length of the tube, and the vibration frequency of the fork, is given by '='-^ I « where x is some integer. Ch. vtti. { 29.] Acoustics, 229 This gives a series of lengths of the resonance tube, any two consecutive ones differing by vl2n. Now vin is the wave-length in air of the note of the fork. So that with a tube of sufficient length, a series of positions of maximum resonance can be determined, the difference between successive positions being half the wave- length in air of the note of the fork. Introducing the correction for the open end of the pipe, the formula (3) for determining the velocity of sound be- comes .4«(/+r) [The most suitable diameter of the tube for a 256 fork is about 5 centimetres ; for higher forks the diameter should be less.] Expejtfnent. — Determine the lengths of the columns of air corresponding to successive positions of maximum resonance for the given fork and deduce the velocity of sound in air. Enter results thus : — Vibration frequency of fork, 256 per sec Lengths of resonance columns : (i) Mean of twelve observations, 31 cm. (2) w » f» 97 n Radius of tube, 2*5 cm. Velocity of sound, from (i) 34,340 cm. per sec. „ „ from (2) 34,000 cm. „ 3a ▼erlficatlon of the Laws of Vibration of Strings. Detennination of the Absolute Fitch of a Note by the Honochord. The vibration of a string stretched between two points depends upon the reflection at either end of the wave motion transmitted along the string. If a succession of waves travel along the string, each wave will in turn be reflected at the one end and travel back along the string and be 230 Practical Physics. [Ch. VIIL § 3a reflected again at the other end ; the motion of any point of the string is, accordingly, the resultant of the motions dne to waves travelling in both directions. Premising that a node Is a point in the string at which the resultant effect of the Incident and reflected waves is to produce no change of posi- tion, and that a loop is a point at which the change of posi- tion due to the same cause is a maximum, it is evident that if a string is to remain in a state of vibration the two ends of the string which are fixed to the supports must be nodes, and it follows that the modes of vibration of the string must be such that the distance between the two ends contains an exact multiple of half the length of a wave, as transmitted along a uniform string of indefinite length and without obstacles. It is shewn in works on acoustics ' that a wave of any length travels along such a string with a velocity v where z^=>/f7«r, T being the stretching force of the string in dynes, and m the mass of a unit of its length expressed in grammes per centimetre. If T be the time of vibration of the note, and X its wave length in centimetres, we have, just as in the case of air, If n be the vibration frequency of the note I r hence X=?=i /Z n n\/ m The distance / between the fixed ends of the string being an exact multiple of -, we have 2 where x is some integer. » Sec Lord Rayleigh's Sounds vol. L chap. vi. ; Thomson and Tart. Elements of Natural Philosophy, Appendix h, p. 284, Ch. viii. § 30.] Acoustics. 231 Whence *~i7V« ^'^ It is this formula whose experimental verification we pro- ceed to describe. The apparatus usually employed for the purpose is known as a monochord or sonometer, and con- sists of a long wooden box with a wire, fixed at one end and stretched between two bridges by a spring at the other, or by means of a weight hanging down over a pulley. The one bridge is fixed at the fixed end of the string ; the other one is movable along a graduated scale, so that the length of the vibrating portion of the string can be read off at pleasiure. The measurement of the stretching force t, either by the hanging weight or by the stretching of a spring attached to the end of the box, is rendered difficult in con- sequence of the fiiction of the bridge, and therefore requires some care. The pulley itself may be used instead of the bridge if care be taken about the measurement of length. For a fine brass or steel wire a stretching force equivalent to the weight of firom 10 to 20 kilogrammes may be employed. This must be expressed in dynes by the multiplication of the number of grammes by 981. It is convenient to have two strings stretched on the same box, one of which can be simply tuned into unison with the adjustable string at its maximum length by an ordinary tuning-key, and used to give a reference note. The tuning can be done by ear after some practice. When the strings are accurately tuned to unison, the one vibrating will set the other in strong vibration also ; this property may be used as a test of the accuracy of tuning. We shall call the second the auxiliary string. It is advisable to use metallic strings, as the pitch of the note they give changes less firom time to time than is the^ case with gut strings. Referring to the formula (i), we see that the note as 232 Practical Physics, [Ch. viii. § 30 there defined may be any one of a whole series, since x may have any integral value. We get different notes on putting X equal to i, 2, 3 ... . successively. These notes may in fact all be sounded on the same string at the same time, their vibration numbers being », 2^ 3», 4» . . . . and their wave-lengths 2/, /, 2/73, 2//4 . . . respectively. The lowest of these is called the fundamental note of the string, and the others har- monics. These may be shewn to exist when the string is bowed, by damping the string — touching it lightly with the finger — at suitable points. Thus, to shew the existence of the first harmonic whose wave-length b /, bow the string at one quarter of its length from one end, and touch it lightly at the middle point The fundamental note will be stopped, and the octave will be heard, thus agreeing in pitch with the first of the series of harmonics given above. To obtain the second harmonic bow the string about one-sixth of its length from the end, and touch it lightly with the finger at one-third of its length. This stops all vibrations which have not a node at one third of the length, and hence the lowest note heard will be the second har- monic, which will be found to be at an interval of a fifth from the first harmonic or of an octave and a fifth fh)m the fundamental tone. We may proceed in this way for any of the series of harmonics, remembering that when the string is damped at any point only those notes will sound that have a node there, and on the other hand, there cannot be a node at the place where the string is bowed ; hence the place for bowing and the place for damping must not be in corresponding positions in different similar sections of the wave-curve ; if they were in such corresponding positions the damping would suppress the vibration of the string alto- gether. The intervals here mentioned may be estimated by ear, or compared with similar intervals sounded on the piano or harmonium. Ch. vin. { 3a] Acoustics, 233 We shall now confine onr attention to the fandamental note of the string. Putting x=si in formula (i) we get *'^'ti\/l <')• We have first to verify that the vibration number of the note varies inversely as the length of the string when the tension is constant This may be done by sliding the movable bridge until the note sounded is at a definite interval from the note of the auxiliary string, with which it was previously in unison. Suppose it to be the octave, then the length of the adjustable string will be found to be one half of its original length ; if a fifth, the ratio of its new length to its original length will be 2/3, and so on ; in every case the ratio of the present and original lengths of the string will be the inverse ratio of the interval In a similar manner we may verify that the vibration frequency varies as the square root of the tension. By loading the scale pan hung from the pulley, until the octave is reached, the load^ will be found to be increased in the ratio of 4 : I, and when the fifth is obtained the load will be to the original load in the ratio of 9 : 4. It yet remains to verify that the vibration frequency varies inversely as the square root of m^ the mass per unit of length of the string. For this purpose the string must be taken off and a known length weighed. It must then be replaced by another string of different material or thickness, the weight of a known length of which has also been deter- mined. Compare then the length of the two strings re- quired to give the same note, that is, so that each is in turn in unison with the auxiliary string. It will be found that these lengths are inversely proportional to the square root of the masses per unit of length, and having already proved that the lengths are inversely proportional to the vibration frequencies, we can infer that the vibration frequencies are ' In estimating the load the weight of the pan must be included. 234 Practical Physics, [Ch. VIIl. § 3a inversely proportional to the square roots of the masses per unit of length. We can also use the monochord to determme the pitch of a note, that of a fork for mstance. The string has first to be tuned, by adjusting the length, or the tension, until it is in unison with the fork. A littie practice will enable the observer to do this, and when unison has been obtained the fork will throw the string mto strong vibration when sounded in the neighbourhood Care must be taken to make sure that the fork is in unison with the fundamental note and not one of the harmonics. The length of the string can then be measured in centimetres, and the stretching force in dynes, and by marking two points on the wire and weighing an equal length of exactly similar wire, the mass per unit of length can be determined Then substituting in formula (2) we get n. This method of determining the pitch of a fork is not susceptible of very great accuracy in consequence of the variation in the pitch of the note of the string, due to altera- tions of temperature and other causes. Experiment — Verify the laws of vibration of a string with the given wire and determine the pitch of the given fork. Enter results thus : — Length of wire sounding in unison with the given fork, 63*5 cm. Stretching force (50 lbs.), 22,680 grammes weight ■» 22680 X 981 dynes. Mass of 25 cm. of wire, '670 grammes. Vibration frequency of fork, 227 per sec 31. Determination of the Wave-length of a high Note in Air by means of a Sensitive Flame. (Lord Rayleigh, Acoustical Observations, PhiL Mag,^ March, 1879.) For this experiment a note of very high pitch is re- quired Probably a veiy high organ-pipe or whistle might Ch. VIII. { 31.] Acoustics, 23s be employedi but a simple and convenient arrangement, the same in principle as a 'bird-call,' consists of two small parallel metallic discs, fixed so as to be a short distance — a millimetre more or less — apart, and perforated, each with a small circular hole the one behind the other. This pair of discs is then fixed on to the end of a supply-tube, and air blown through the holes by means of a loaded gas-bag or bellows. It is convenient to connect a mano- meter with the supply-tube, close to the whistle, in order to regulate the supply of air firom the reservoir, and thus maintain a note of constant pitch. Fig. 18 shews a section of this part of the apparatus. It is very easily constructed The one disc can be fixed to the tube of glass or metal by sealing wax, and the other adjusted and kept in its place with soft wax. A sensitive gas fiame •flares* when a note of sufiiciently high pitch is sounded in its neighbour- hood ; thus a hiss, or the shaking of a bunch of keys is generally effective. To obtain a sensitive flame, a Fig. 18. I v^ pin-hole steatite burner may be employed; it must be supplied with gas at a high pressure (9 or 10 inches of water) from a gas holder. The ordinary gas supply of a town, which gives only about i inch pressure, is of no use for the purpose. The tap — best an india-rubber tube with pinch-cock — which regulates the flame, must be turned on until the flame is bumbg steadily (it will generally be some 18 inches high), but just on the point of flaring. The sound of the * bird-call,* described above, will then, if it be high enou>2jb^ 236 Practical Physics. [Ch. VIIT. § 3«« make the flame flare, but it will recover its steadiness when the sound ceases. In order to determine the wave-length of a note by thii apparatus, a board is placed so that the sound is reflected perpendicularly from its surface. Placing the nozzle of the burner in the Hne from the source of sound perpendicular to the board, and moving the burner to and fro along this line, a series of positions can be found in which the effect of the sound upon the flame is a minimum. The positions are well-defined, and their distances from the board can be measured by taking the distances between the board and the orifice of the burner with a pair of com- passes, and referring them to a graduated scale. These positions correspond to the nodal points formed by the joint action of the incident vibration and the vibration refiected from the surface of the board. The distance between consecutive positions corresponds accordingly to half a wave-length of the incident vibration. The wave- length of the note sounded is, therefore, twice the distance between consecutive positions of minimum effect upon the fiame. The distances of as many successive positions as can be accurately observed should be taken. Each observation should be repeated three or four times and the mean taken. Instead of the sensitive flame, an india-rubber tube lead- ing to the ear may be employed, and positions of silence determined. It must be remembered, however, in this case that the position of silence for the ear corresponds to a position of minimum pressure-variation at the orifice of the tube — that is to say, to a loop and not to a node. The distances of these positions of silence from the wall are, therefore, odd multiples of quarter-wave-lengths instead of even multiples, as when the sensitive fiame is used. Experiment— ^tXtxicimt. the wave-length of the given note by means of a sensitive flame. Cm. Vin. § 31.] Acoustics. Enter results thus : — 23; No.orpom. tiooof mini- mum effect, redcomng from the board Actual obtervations of the db- txuice in mm. of the nozxle from the board. Mean of Observations Half.Wave. Length de- duced in Millimetres. _ i6|, 16J, 16, 16 31, 3ii, 32J, Zh 32 47. 47i 46i, 47, 45i 62, ezh 64, 60k 62k 78i. 78J i6'25 315 4675 62*25 785 16-25 1575 15-6 «5-5 Mean wave-length - 31*2 mm. CHAPTER IX. THERMOMBTRY AND EXPANSION. Thb temperature of a body may be defined as its thermal condition, considered with reference to its power of com- municating heat to or receiving heat from other bodies. This definition gives no direction as to how the temperature of a body is to be measured numerically. We may amplify it by saying that i^ when a body a is placed in contact with another body b, heat passes from a to b, the body a is at a higher temperature than b ; but this extension only indi- cates the order in which a scale of temperatures should be arranged. In order to measure temperature we may select one of the effects produced by an accession of heat in a particular instrument, and estimate the range of temperature through which that instrument is raised or lowered when placed in contact with the body whose temperature is to be measured by measuring the amotmt of the effect produced. This is the method practically adopted The instnmient which is 238 Practical Physics. [C\\. ly. so used is called a thermometer, and the branch of the science of heat which treats of the application of such in- struments is called thermometry. A continuous accession of heat produces continuous alteration in many of the physical properties of bodies, and any one of them might have been selected as the basis of a system of thermometry. Attempts, which have met witb more or less success, have been made to utilise several of these continuous alterations for the purpose. The change of volume of various liquids enclosed iii glass vessels ; the change in pressure of a gas when the volume is kept con- stant, or the change in volume when the pressure is kept constant ; the change in the electrical resistance of a wire ; the change in the electromotive force in a thermo-electric circuit ; the change in length of a metallic bar ; the change in the pressure of the vapour of a liquid ; change of shape of a spiral composed of strips of different metals, as in Br^guet's thermometer, have all been thus employed. Of all these methods of forming a system of thermo- metry, the one first mentioned is by far the most frequently employed It owes its general acceptance to the fact that the change of volume of a liquid in a glass vessel is very easily measured with great accuracy. Moreover, if it were not for certain slow-working changes of very small magni- tude in the volume of the glass envelope, of which we shall speak later, the indication of such an instrument would practically depend upon the temperature and upon nothing else. The liquids which have been employed are mercury, alcohol, and ether. Mercury can easily be obtained pure, and remains a liquid, with a vapour- pressure less than the ordi- nary atmospheric pressure for a wide range of temperatures, including those most frequently occurring in practice. Ether has a larger coefficient of expansion, but can only be used for a small range of low temperatures. The thermometers most generally in use are accordingly filled with mercury, and the expansion of mercury in a glass vessel has thus been Ch. IX.] Thermometry and Expansion, 239 adopted as the effect of heat to be employed as the basis of the numerical measurement of temperature. A mercury thermometer consists of a stem, a glass tube of very fine and uniform bore, having a cylindrical or spherical bulb blown at the end. The bulb and part of the tube are filled with mercmy, and the top of the stem is hermetically sealed, when the bulb is so heated that the whole instrument is filled with the liquid. When the mercury cools and contracts, the space above it is left empty. The numerical measurement is introduced by marking upon the stem the points reached by the mercury when the thermometer is maintained successively at each of two temperatures which can be shewn to be constant, and dividing the length of the stem between the two marks into a certain number of equal parts. These two fixed tempera- tures are usually the temperature of melting ice, and the temperature of steam which issues from water boiling under a standard pressure of 760 mm. They have been experi- mentally shewn to be constant, and can always be obtained by simple apparatus (see § 33). The two marks referred to are called the fi-eezing and the boiling point respectively, and the distance between them on the stem is divided into 100 parts for the centigrade thermometer, and 180 for the Fahrenheit, each part being called a degree. On the former the freezing point is marked o**, and on the latter 32®. The remarks which follow, when inappli- cable to both kinds, may be held to refer to the centigrade thermometer. It should first be noticed that this system, which supplies die definition of the numerical measure of temperature, is completely arbitrary. A number of degrees of temperature corresponds to a certain percentage of the total expansion of mercury in a glass vessel between o** and 100**. Two quantities of mercury will doubdess expand by the same fraction of their volume for any given range of temperature. 240 Practical Physics. [Ch. IX, and thus two mercury thermometers, similarly graduated, may be expected to give identical indications at the same temperature, provided each tube is of uniform bore, and the expansion of the glass, as referred to the corresponding expansion of the mercury, is uniform for each instrument This is in general sufficiently nearly the case for two ther- mometers which have been very recently graduated. But a thermometer filled with any other hquid, and agreeing with a mercury thermometer at two points, cannot be expected to, and does not in fact, agree with it for temperatures other than those denoted by the two points. If it did it would shew that the rate of expansion of its liquid in glass was uniform for successive intervals of temperature, as defined by the mercury thermometer, and this is generally not the case. Even the conditions necessary for two mercury thermo- meters to give identical indications at the same temperature are not, as a rule, satisfied. In the first place, the bore of a thermometer is not generally unifonn. The variation may, indeed, be allowed for by cahbration (see § 8), so that we may correct the indications for want of uniformity of bore ; the determination of the corrections in this way is a somewhat tedious operation. Moreover, the volume of the glass envelope undergoes a slow secular change. A thermometer bulb, when blown and allowed to cool, goes on contracting long after the glass has attained its normal tem- perature, the contraction not being quite complete even after the lapse of years. If the bulb be again heated, the same phenomenon of slow contraction is repeated, so that, after a thermometer is filled, the bulb gradually shrinks, forcing the mercury higher up the tube. If the thermometer has been already graduated, the effect of this slow contraction will appear as a gradual rise of the freezing point In some thermometers the error in the freezing point due to this cause amounts to more than half a degree, and the error will affect the readings of all temperatures Ch. IX.l Tliermonietry afiii Expansion. 241 between o** and 100° by nearly the same amount The in- strument should, therefore, not be graduated until some considerable time after being filled; but even when this precaution is taken the change in the zero point is not completely eliminated, but only considerably diminished. A corresponding small change of the zero point is set up whenever the thermometer is raised to the boiling point The reading of a mercury thermometer does not, there- fore, give an indication of temperature which will be clearly understood by persons who do not measure temperatures by that particular thermometer. To ensure the reading being comparable with those of other instruments, the tube must have been calibrated, and the fixed points quite recently re-determined, and the readings thus corrected ; or, adopt- ing another and more usual method, the individual ther- mometer in question may be compared experimentally with some instrument generally accepted as a standard. A set of such are kept at the National Physical Laboratory ; they have been very carefully made and calibrated, and their fixed points are repeatedly determined, and a standard scale is thus established. With one or more of these standards any thermometer can be compared by immersing them in water which is kept well stirred, and taking simultaneous readings of the two at successive intervals of temperature. In this way a table of corrections is formed for the thermometer which is tested, and its indications can be referred to the Kew standard by means of the table. However, the secular contraction of the bulb may still be going on ; but to allow for any contraction subsequent to the Kew com- parison, it is suflficient to ascertain if there has been any change in the freezing point, and in that case consider that an equal change has taken place for every temperature, and that, therefore, each correction on the table is changed by that amount A specimen table of Kew corrections is appended as an 242 Practical Physics. [Ch. ix. example of the way in which this method of referring ther- mometers to a common standard is worked. Thxr. Fobm. D. KEW OBSERVATORY.— Certificate of ExaminatioiL Centigrade THERMOMETER.-NO. /^^/S ^^^^ ^c/^-^. by ^. t^u>n^, l£o9UMn. (VERIFIED UNMOUNTED AND IN A VERTICAL POSITION.) CmrecHons to he applied to ike Scale Readings, determined by eomparison with the Standard Instruments at the Ktw Observatory. o o At O -Oy 5 -<>-^ lo -0*y 15 -0-y 20 -0«« 25 -o-« 30 — 0'« 35 -^-4 Ar#/#— L— When the sign of the Conreaion is +, the quantity is to be added \o iJ-e observed Kading, and when — to be *«3/ra<:/^<£ from it. II.— Mercurial Thermometers are liable, through age, to read too high ; this instrument ought, therefore, at some future date, to be again tested at the melt* ing point of ice, and if its reading at that point be found different from the one now given, an appropriate correction should be applied to all the above points. Kew Observatory, Superintendent. MST. 500— s 78. So far we have dealt with the principles of the method of measuring temperatures within the range included between the freezing and boiling points of water. In order to extend the measurement beyond these limits various plans have Ch. IX.] Thermometry and Expansion, 243 been adopted. The mercury thermometer is sometimes used, its stem beyond the limits being divided into degrees equal in length to those within the limits. A thermometer divided in this way can be used for temperatures down to — 40**, and up to 350° C ; but, unfortunately, the difference in the expansion of different specimens of glass is such that at the higher temperatures two thermometers, similarly gra- duated, may differ by as much as ten degrees, and hence the mercury thermometer thus used does not give a satisfactory standard. Two air thermometers, on the other hand, when properly corrected for the expansion of the glass, always give the same readings, and thus the air thermometer has come to be recognised as the temperature standard for high and low temperatures. It is referred to the mercury standard for the freezing and boiling points and intermediate tem- peratures; thus the higher temperatures are expressed in centigrade degrees by a species of extrapolation, using the formula for the expansion of a permanent gas as deter- mined by observations within the limits of the mercury thermometric standard Other methods of extrapolation from a formula verified by comparison, either with the mercury or air thermometer, have sometimes been employed with more or less success, in order to determine temperatures so high that the air thermometer is unsuitable, such as, for instance, the tem- perature of a furnace. In the case of Siemens' resistance pyrometer, a formula is obtained by experiments at low temperatures, expressing the relation between the resistance of a platinum wire and its temperature ; the temperature of the fiimace is then deduced from an observation of the resistance of the platinum on the supposition that the formula holds, although the temperature is a long way out- side the limits of verification. The temperature obtained in some manner, generally analogous to this, is often ex- pressed as so many degrees centigrade or Fahrenheit It is evident that numbers obtained by different methods may R 2 244 Practical Physics. [Chap. IX. be widely different, as all are arbitrary. At present it is a matter of congratulation if two different instruments on the same principle give comparable results ; and, until some more scientific, or rather, less arbitrary, method of measuring temperatures is introduced, the precise numbers quoted for such temperatures as those of melting silver or platinum must remain understood only with reference to the particulai system of extrapolation adopted to extend the range of numbers from those properly included in the range of the mercury thermometer, namely, those between the freezing and boiling points of water. 32. Constmotion of a Water Thermometer. The method of filling a thermometer is given in full in Deschaners^ or Ganot's 'Natural Philosophy,' and Maxwell's *Heat.' In this case water is to be used instead of mercmy. One or two points may be noticed : — (i) The tube and bulb have not always a cup at the top. When this is the case, a piece of wide glass tubing must be drawn out to serve as a funnel, and joined by means of clean india-rubber to the tube of the thermometer. (2) It would be difficult to seal the glass tube when fiill of water, unless it has been previously prepared for closing. After the bulb has been filled, but before it is again heated to the high temperature, the upper part of the tube is softened in a blow-pipe flame, and drawn out so as to leave a fine neck in the tube. Then the bulb is heated until the liquid rises above this neck, and when this is the case the tube is sealed by applying a small blow-pipe fiame at the thinnest part At the moment of seahng the source of heat must be removed firom the bulb, otherwise the liquid will continue to expand, owing to the rise of temperature, and will burst ffe'ai ^^^^*^^"^^' Natural Philosophy, p. 245, etc. 5 see also Preston's Ch. IX. 5 32.] TJiermometry and Expansion. 245 the bulb. The safest way of heating the bulb is to put it in a bath of liquid — melted paraffin, for example, or water if the thermometer be not required for use near the boiling point — and apply heat to the bath until the hquid in the thermo- meter reaches beyond the neck. Remove the source of heat from the bath and seal off the tube as the level of the water sinks past the narrow neck. (3) The water used for filling the thermometer should be distilled water from which the dissolved air has been driven by long-continued boiling. This precaution is essential, as otherwise bubbles of air separate from the water in the bulb and stem after sealing, and this oflen renders the thermometer useless until it has been unsealed and the air removed and the tube re-sealed. We proceed to shew how to use the thermometer to de- termine the variation of volume of the water. We require, for this purpose, to know the volume of any given length of the tube and the whole volume of water con- tained in the thermometer. To find the Volume of any Length of the Tube, Before filling the thermometer, introduce into the tube a small pellet of mercury and measure its length, which should be from 10 to 20 cm. Then warm the bulb and force the mercury out into a beaker, of which the weight is known. Weigh the beaker and mercury, and get by subtraction the weight of the mercury. Now, we may take the density of mercury to be 13*6. If, then, we divide the mass in grammes by this number, we get the volume in cubic centimetres* We thus find the volume of a known length — that of the pellet of mercury — of the tube, and from this can determine the volume of any required length. For greater accuracy it is necessary to measure the length of the same pellet of mercury at different parts of the tube, thus calibrating the tube (see § 8). To find the Volume of the Water which is contained in tlu Thennorneter, Weigh the bulb and tube when empty, then weigh it again 246 Practical Physics. [Ch. IX. § 32. when filled, before sealing off. The difference in the weights gives the number of grammes of water in the bulb and tube, and hence the number of cubic centimetres of water in the two can be calculated. It may be more convenient to seal off before weighing, but in this case great care must be taken not to lose any of the glass in the act of sealing, and to put the piece of glass which is drawn off on the balance with the tube. Let us suppose the volume of i cm. length of the tube is *oi C.C, and that the volume of the water contained is 4-487 cc After sealing the tube as already described, immerse it in a bath of water at the temperature of the room, noting that temperature by means of a thermometer ; suppose it to be i5*» C Make a series of marks on the tube at a known distance above the level of the water in it ; let us say at each centi- metre. Now raise the temperature of the bath until the level of the water in the tube rises to these marks successively, and note the successive temperatures as indicated by the other thermometer. In this way determine the temperatures corresponding to the successive steps in the expansion of the water estimated in fhictions of the original volume. Set out in a diagram points representing these temperatures and the corresponding volumes in the manner suggested in chap, iii., pp. 50, 51. A continuous curve can then be drawn passing through the series of points, and the curve so drawn will represent approximately the true course of the variation of volume with temperatiure, of the water relatively to glass. From it, the mean coefficient of relative expansion for any interval can be determined by dividing the change of volume during the interval by the change of temperature, and the true relative coefficient at any temperature can be inferred from the tangent of the angle which the tangent to the curve (at the point corresponding to that temperature) Ch. IX. § 32.] T/iermotneiry and Expansion, 247 makes with the temperature-axis after the same manner as velocity is inferred in § b. chap, v.* p. 147. Experiment.^Dtitrmwit by means of a water thermometer the thermal expansion of water relative to glass. Enter results thus : — Length of pellet of mercury 15*3 cm. Weight of do. 2*082 gm. VoL of I cm. of tube 'oi cc VoL water initially 4487 cc Temp. 15** VoL finally 4-587 cc. Temp. 70** Mean coeff. of expansion - -000405 per 1°. 33. Thermometer Testing. By this we mean determining the indications of the thermometer which correspond to the freezing point of water and to its boiling point under a pressure of 760 mm. The first observation is made by placing the thermo- meter so that its bulb and stem up to the zero are sur- rounded with pounded ice. The ice must be very finely ];>ounded and well washed to make quite sure that there is no trace of salt mixed with it. This precaution is very im- portant, as it is not unusual to find a certain amount of salt with the ice, and a very small amount will considerably re- duce the temperature. The ice should be contained in a copper or glass funnel in order that the water may run off as it forms. The ther- mometer should be supported in a clip, lest when the ice melts it should fall and break. The boiling point at the atmospheric pressure for the time being may be determined by means of the hypsometer, an instrument described in any book on physics.^ The thermometer to be tested must be passed through the cork at the top of the hypsometer, and there fixed * Garnetl, Heat, § 12, &c. Deschanel, Natural Philosophy ^ p. 248, &c 248 Practical Physics. [Ch. IX. f 33. so that the 100° graduation is just above the cork. One aoerture at the bottom of the cover of the hypsometer is to allow the steam issuing from the boiling water to es- cape ; to the other aperture is attached by an india-rubber tube a pressure gauge, which consists of a U-shaped glass tube containing some coloured liquid. The object of this is to make sure that the pressure of the steam within the hypsometer is not greater than the atmospheric pressure. The water in the hypsometer must be made to boil and the thermometer kept in the steam until its indication becomes stationary. The temperature is then read. In each of these operations, in order to make certain of avoiding an error of parallax in reading (i.e. an error due to the fact that since the object to be read and the scale on which to read it are in different planes, the reading will be different according as the eye looks perpendicularly on the stem or not), the thermometer must be read over the edge of a card or by a telescope at the same height as the graduation to be read. If, then, the thermometer be vertical, the line of sight being horizontal will be perpen- dicular to it (It must be remembered in esrimating a fraction of a division of the thermometer that in the telescope the image of the scale is inverted.) We thus determine the boiling point at the atmospheric pressure for the time being. We have still to correct for the difference between that pressure and the standard pressure of 760 mm. To do this the height of the barometer must be read and expressed in millimetres. We obtain from a table shewing the boiling point for different pressures, the fact that the difference in the temperature of the boiling point of i*» corresponds to a difference of pressure of 26*8 mm. We can, therefore, calculate the effect of the difference of pressure in our case. Suppose the observed boiling point reading is 99-5, and the height of the barometer 752 mm. We may assume'that for small differences of pressure from the standard pressure,' Ch. IX. § 33.] Thermometry and Expansion. 249 the difTerence in the boiling point is proportional to the difTerence of pressure ; hence 760—752 required correction. 26-8 I o 8* ••. the required correction = —^^ = '3*. ^ 26*8 And therefore the corrected boiling point would read 99 '8** on the thermometer. The correction is to be added to the apparent boiling- point leading if the atmospheric pressure is below the standard, and vice vers&. The difference of temperature of the two boiling points depends only on the difference of pressure. Also an increase of pressure of i mm. of mercury produces an alteration of the temperature of the boiling point of 0-0373° C, or an increase of temperature of the boiling point of 1° corresponds to a pressure of 26*8 mm. of mercury. Now the specific gravity of mercury referred to water is 13*6, that of dry air at 760 mm. pressiure, and 15® C. temperature is '001225. Thus the pressure due to i mm. 1 1*6 of mercury is equal to that due to — ^- mm., or 11*102 ' '001225 metres of dry air. But a rise in temperature of 1° corresponds to an increase in pressure of 26-8 mm. mercury ; that is, to an increase of pressure due to ii'io2 X26-8 metres of dry air. Thus, the boiling point alters by i** C. for an alteration of pressure equal to that due to a column of dry air at 15** C and of 297*5 metres in height Experiments. (i) Determine the freezing and boiling points of the given thermometer. Enter results thus :— Thermometer, Hicks, No. 14459. Freezing point -o**'!. Boiling point 99""'^ 250 Practical Physics. [Ch. IX. § 33 The following additional experiments may be performed with the hypsometer. (2) Put some salt into the hypsometer and observe the boiling point again. (3) Tie some cotton wick round the bulb pf the thermometer, and let the end drop into the solution. Vide Gamett, § 13. (The cotton wick should be freed from grease by being boiled in a very dilute solution of caustic potash and weU washed.) (4) Remove the water, clean the thermometer, and repeat the observation with a given liquid. Boiling point of alcohol is 79®. „ „ ether 37**. „ „ turpentine 130^ (5) Clean the thermometer and hypsometer, and remove the apparatus to a room in the basement, and observe the temperature of the boiling point of water. Take the apparatus up to the top of the building and repeat, and from the two observations determine the height of the building. 34. Boiling Point of a Liquid. A liquid is usually said to boil at a temperature t when the pressiure of its vapour at this temperature is equal to the external pressure/. But if the sides of the vessel be smooth and die liquid be quite free from dissolved air, or if it contain salts in solution, it will generally not boil till its temperature is higher than /. Suppose the liquid to boil at /^'-fr, then the vapour rising up at this temperature will exert a pressure greater than the external pressure /. Consequently it will expand till its pressure falls to /, its temperature at the same time falling till it reaches the corresponding temperature /.* Hence the temperature of the vapour over a boiling liquid under a given pressure /, fa a constant quantity under all > Maxwell, ffeat^ pp. 25 and 289. Ch. IX, 5 34.] TItemtonutry and Expansion. 251 circumstances, and is called the boiling point of the liquid under the pressure /. The hypsometer will serve to determine the boiling point of a liquid. In many cases, however, when the quantity of liquid obtainable is small, the apparatus described below is more convenient. The liquid is put into the outer glass tube (a). The inner tube (b), made of brass, is then restored to its fk. .9. place, as in fig. 1 7, and the whole placed on a sand bath and heated by a Bunsen burner. When the Hquid boils, the vapour will enter by the aperture o into the tube b, and will leave b by the glass tube d, which should be connected by a short piece of india-rubber tube with a condenser, to prevent the vapour entering the room. As the boiling continues, the thermo- meter will rise at first, but afterwards remain stationary. Enter this reading, and also the height 0/ the barometer at the same time. 35. FaBing Foint of a Solid The method to be adopted in order to determine the fuang point of a solid must depend on several considera- tions, as — (1) Whether the temperature can be registered on a mercuiy thermometer; i.e; docs it lie between —40° C. and +350° c.? (2) Does the solid pass direcdy from the solid to the liquid state, or is there an intermediate viscous condition ? If so, the melting point may be taken as somewhere between the temperature of the liquid and solid condition, but cannot be considered as a definite temperature. (3) Whether or not the subsunce is a good conductor of heat If it h>e, the temperature of a vessel containing the tubstance in part solid will be veiy neatly constant if kept 252 Practical Physics. [Ch. IX. § 35- properly stirred This is the case with ice and the fusible metals and alloys. For bodies which are bad conductors a method has to be adopted as occasion requures. We give as an instance the following, which is available in the case ot paraffin wax. The thermometer, when dipped into* the melted paraffin, is wetted by the liquid, and when taken out is in con- sequence covered with a very thin and perfectly transparent film of liquid paraffin. This film cools, and on solidifying assumes a firosted appearance which extends rapidly all over the part of the thermometer that has been immersed. If the bulb of the thermometer is sufficiently small for us to neglect the difference of temperature between the interior and exterior portions of the mercury, the observation of the thermometer at the instant when this frosted appearance comes over the bulb may be taken as the melting point of paraffin. The only error likely to be introduced is that mentioned above, viz. that the temperature of the parafi^ is not the mean temperature of the thermometer bulb. This can be rendered smaller and smaller by taking the liquid at temperatures approaching more and more nearly to the melting point as thus determined, and its direction can be reversed if we allow the paraffin to solidify on the bulb and then heat the bulb in a beaker of water and note the tem- perature at the instant when the film becomes transparent The mean of this temperature and that deduced from the previous experiment wiU be the melting point. The following is another method of obtaining the fusing point of a solid such as paraffin. Draw out a fine glass a beaker of wate. ^d W kl^rr^^^^ ^^ ^^^ ^ Uirrcd. When tb «^1 ffin 1 T I' ^^^'"^ '^^ ^^^ ^^" n ui . paraffin melts it becomes transparent, and Cii. IX. § K..] Tlurinomelry and Expansion. 253 the temperature at vhich this change takes place can be noted with considerable accuracy. Raise the temperature a little above the melting point, and allow the bath to cool slowly. When the paraffin solidifies it becomes opaque. By alternately heating and cooling within narrow limits, a series of values of the melting point which only diiler very little can be obtained ; the mean of these may be taken as the melting point of paraffin, K. Effect of DiisolTsd Salts on the Freezing Point. For the more accurate determination of the freezing poini of a solution, and of the effect fig. uL of dissolved salts in altering the freezing point, the following apparatus, described by Beek- man, may be used. The glass tube a (fig. xxi) c»3ntain8 a delicate thennometer T, 3 stirrer of stout platinum wire, and the liquid to beexperi- mented on. The salt whose effect it is wished to study can be introduced by a side tube u, sealed on to a, or more simply through a glass tube passing through the cork which closes the upper end of a. The tube A is placed inside a wider tube C passing through a cork in the open end of c. This tube merely serves as an air-jacket c passes through the lid of a wide glass vessel d, which con- tains water or a freezing mix- ture, the temperature of which should be some 5° below the freezing point of the liquid in a. The bath also contains 254 Practical Physics, [Ch. IX. 5 k. a stirrer. A weighed quantity of liquid is placed in a and the whole allowed to cool slowly, being kept at the same time well stirred. The liquid in a is probably thus cooled below its freezing point, freezing then takes place, and the thermometer rises suddenly to the melting point as the solid separates out The freezing point of the solvent — water, or whatever it may be that separates out on freezing — is thus determined. A known quantity of the substance whose effect is required is introduced through the side tube b, and the experiment is repeated. The effect of the salt in modifying the freezing point of the solvent is thus found. It has been shewn by Raoult and others that, for a large number of substances, when a mass of the substance^ p grammes, is dissolved in a solvent, the mass of the solution being w^ then the product of the molecular weight of the substance multiplied by the depression of the freezing point is proportional topjw^ so that, if m be the molecular weight of the salt, A the depression of the freezing point, and k a constant, then for a large class of salts »j A = kpjw. This law may be verified by finding the depression of the freezing point produced by the addition of various amounts of the same salt, and then by comparing the de* pression produced by dissolving equal quantities of different salts so as to form solutions of equal volumes. The quantity k is the product of the molecular weight of the salt, the depression of the freezing point produced by the solution of one gramme of salt and the mass of the solution. The results are generally stated on the supposition that m grammes of salt are dissolved per litre of the solutioa We then have p-=^m^ w = mass of one litre of solution ; w A is the depression produced when m grammes of salt {m being the molecular weight) are dissolved in i litre of Ch. IX, ] Thermometry and Expansion, 255 the solvent Thus, according to the law, A should be con- stant. It 48 found to have the value i-8** C. approximately. According to Raoult there is a relation between the constant k^ the absolute temperature of solidification t, and the latent heat l. By supposing a small quantity of the liquid taken round a thermodynamic cycle at the tem- perature of solidification, he shews that k = 2T*/l. This result may be verified if the latent heat of fusion of the substance be known. Experiment, — Shew that the lowering of the freezing point of a solvent due to the presence of a salt is proportional to the mass of salt dissolved, and inversely proportional to its mole- cular weight. COEFFICIENTS OF EXPANSION. For any ordinary substance, with the exception of water, the changes of volume for equal increments of temperature are so nearly equal that the expansion may be calculated from a coefiScient approximately constant for each substance, which may be defined as follows : — Definition, — A coefficient of expansion by heat may be defined as the ratio of the change of a volume, area, or length per degree of temperature to the value of that volume, area, or length at zero centigrade. In solids and liquids the expansion is so small that in practice we may generally use^ instead of the value of the quantity at zero, its value at the lower of the two tempera- tures observed in the experiment For solid bodies we have the coefficients of linear, super- ficial, and cubical expansion depending on the alteration of length, breadth, or thickness (linear), of surface (superficial), and of volume (cubical) respectively. Let a, /3, 7 be these three respectively, and suppose the body to be isotropic, i.e. to have similar properties in all directions roimd any given point; then it can be shewn that /3=2a, 7=30. 256 Practical Physics. fCw- DC For consider a rectangle the sides of which are a and ^. When the temperature is raised by f* the sides increase respectively by a a / and 3 a /, so that their new values are fl(i+a/) and 3(1 -l-a/). Thus the area is a^(i + a /)*, or, since a is very small, tf3(i + 2o/). But if /3 be the co- efficient of superficial expansion, the new area is <i^(i +/3/). Thus we have )3=2 a. In a similar way considering the expansion of a cube we may shew that 7=3 a. For liquid bodies we have to deal only with the coefficient of cubical expansion. Any measurement of expansion is attended with con- siderable difficulty. A liquid requires to be contained in some vessel, and thus we have to consider the alteration in volume of the vessel as well as that of the liquid itself. In the case of a solid, any cause which changes the temperature of the body to be measured probably changes that of the measuring appa- ratus and causes it to expand also. Our measurements will therefore give the expansion of one substance relatively to another. Thus, we should find, mercury and most liquids expand considerably as compared with glass, while the metals expand greatly in comparison with wood or stone. Methods, it is true, have been devised for determining the absolute expansion either of a liquid or a solid, but tl ese are too complicated for an elementary course. We shall explain how to determine (i) by means of read- ing microscopes, the coefficient of linear expansion of any solid which can be obtained in the form of a long rod, and (2), by means of the weight thermometer, the coefficient of expansion of a liquid and also that of cubical expansion ot a solid. In the case of a gas we may consider either the altera- tion of volume under constant pressure or the alteration of pressure at constant volume. We shall describe experi- mental methods of measuring these two. Ch. DL § j6b] Thermometry and Expansion. 2$? 56. Coefficient of Linear Ezpaniion of a Bod. We require to measure the length of a rod, or the dis- tance between two marks on it, at two known temperatures, say 15** C and 100** C. The highest degree of accuraqr requires complicated apparatus. The following method is simple, and will give very fair results. A thick straight rod is taken, about 50 cm. in length, and a glass tube of 4 or 5 cm. bore and somewhat greater length than the rod. The tube is dosed with a cork at each end, and through each cork a small piece of glass tubing is passed, and also a thermometer. Two fine scratches are made on the rod, one close to each end, at right angles to its length, and two other scratches, one across each of the former, parallel to the length. The glass tube is damped in a horizontal position and the rod placed inside it, resting on two pieces of cork or wood in such a manner that the scratches are on the upper surface and can be seen through the glass. The whole should rest on a large stone slab — a stone wmdow-siU serves admirably. The piece of glass tubing in one of the corks b connected with a boiler from which steam can be passed into the tube, the other communicates with an arrangement for condensing the waste steam. A pair of reading microscopes are then brought to view the cross-marks on the rod, and are clamped securely to the scone. The microscopes, described in § 5, should be placed so that they slide parallel to the length of the rod ; this can be done by eye with sufficient accuracy for the purpose. If microscopes mounted as in § 5 are not available, a pair with micrometer eye-pieces, or with micrometer scales in the eye-pieces, may be used. For convenience of focussing on the rod which is in the glass tube, the microscopes must not be of too high a power. Their supports should be clamped down to the stone at 258 Practical Physics, (Ch. IX. § 361 points directly behind or in front of the position of the microscopes themselves, to avoid the error due to the ex- pansion of the metal slides of the microscopes, owing to change of temperature during the experiment Call the microscopes a and b ; let a be the left-hand one of the two, and suppose the scale reads from left to right. Turn each microscope-tube round its axis until one of the cross-wires in the eye-piece is at right angles to the length of the rod, and set the microscope by means of the screw until this cross-wire passes through the centre of the cross on the rod. Read the temperature, and the scale and screw-head of each microscope, repeating several times. Let the mean result of the readings be Temp. A B 15* C . . . 5'io6 cm, 4738 cm. Now allow the steam to pass through for some time ; the marks on the copper rod will appear to move under the microscopes, and after a time will come to rest again. Follow them with the cross- wires of the microscopes and read again. Let the mean of the readings be Temp. A B 100* C . . . 5*074 cm. 4780 cm. Then the length of the rod has apparently increased by 5-106- j'o74-f478o-4 738, or -074 cm. The steam will condense on the glass of the tube which surrounds the rod, and a drop may form just over the cross and hide it from view. If this be the case, heat from a smaD spirit flame or Bunsen burner must be applied to the glass in the neighbourhood of the drop, thus raising the tempera- ture locally and causing evaporation there. Of course the heating of the rod and tube produces some alteration in the temperature of the stone slab and causes it to expand slightly, thus producing error. This will be very slight, and for our purpose negligible, for the rise of Cm. IX. § 3d.] Thermometry and Expansion, 259 temperature will be small and the coefficient of expansion of the stone is also smalL We have thus obtained the increase of length of the rod due to the rise of temperature of 85°. We require also its original length. To find this, remove the rod and tube and replace them by a scale of centimetres, bringing it into focus. Bring the cross-wires over two divisions of the scale, say 10 and 6o« and let the readings be A B 4*576 cm. 5'2i3 cni. Then clearly the length of the rod at 15"* is So-(5io6-4-576) + (4738-5*2i3X or 48-995 cm. To find the coefficient of expansion we require to know the length at o® C; this will differ so little from the above that we may use either with all the accuracy we need, and the required coefficient is - — ^^ — , or -0000178. ^ 85x48-995 Experiment. — Determine the coefficient of expansion of the given rod. Enter results thus : — Increase of length of rod between 15** and 100® -074 cni. Length at 15® 48995 cm. Coefficient ....... -0000178 37. The Weight Thermometer. The weight thermometer,* consists of a glass tube closed at one end, drawn out to a fine neck, which is bent so that it can easily dip into a vessel of liquid. It is used (i) to determine the coefficient of expansion of a liquid relatively to glass ; (2) to determine the coefficient of expansion of a solid, that of the liquid being known. • Garnett, Htat^ §§ 80, 84. Dcschanci, Natural Philosophy^ p. 283. s 2 26o Practical Physics. [Ch. IX. § jie. For (i) we fir^t fill the thermometer with the liquid and determine the weight of liquid inside, when the whole is at some known low temperature, e.g. that of the room or that of melting ice. We then raise the thermometer and liquid to some higher temperature, that of boiUng water, sup- pose. Part of the liquid escapes from the open end. The weight of that which remains inside is then determined, and from these two weights, and the known difference between the temperatures at which they respectively fill the thermo- meter, we can calculate the coefficient of expansion of the liquid relatively to the glass. Our first operation will be to weigh the empty glass tube, which must be perfectly clean and dry. Let its weight be 5*621 grammes. We now require to fill it For this purpose it is heated gently in a Bunsen burner or spirit lamp, being held during the operation in a test-tube holder. Its neck is then dipped under the surface of the liquid whose coefficient of expansion is required — glycerine, suppose— and the tube allowed to cooL The pressure of the external air forces some of the glycerine into the tube. As soon as the liquid ceases to run in, the operation is repeated, and so on imtil the tube is nearly full. It is then held with its orifice under the glycerine, and heated until the fluid in the tube boils. The air which remained in is carried out with the glycerine vapour and the tube left filled with hot glycerine and its vapour. The flame is removed and the thermometer again cooled down, when the vapour inside condenses and more liquid is forced in by the external air pressure. If a bubble of air is still left inside, the operation of heating and cooling must be repeated until the bubble is sufficiently snudl to be got rid of by tilting the thermometer so that it floats up into the neck. There is another plan which may sometimes be adopted with advantage for partially filling the thermometer. Ch. IX. § 37.] Thermometry and Expansion, 261 Place it, with its beak dipping into the glycerine, undei the receiver of an air-pump and exhaust The air is drawn both out of the thermometer and the receiver. Re-admit the air into the receiver. Its pressure on the surface of the glycerine forces the liquid into the tube. It is difficult, however, by this method to get rid of the last trace of air. Suppose the thermometer is filled ; it is now probably considerably hotter than the rest of the room. Hold it with its beak still below the surface of the glycerine and bring up to it a beaker of cold water, so as to surround with water the body of the tube and as much as possible of the neck. This of course must not be done too suddenly lest the glass should crack. Let the thermometer rest in the beaker of water — its orifice still being below the surface of the glycerine — and stir the water about, noting its temperature with an ordinary thermometer. At first the temperature of the water may rise a little ; after a time it wiU become steady, and the tube may be removed. Let the observed temperature be 15** C. We have now got the weight thermometer filled with glycerine at a temperature of 15° C Wei^ the tube and glycerine ; let the weight be 16*843 grammes. The weight of glycerine inside then is 16*843 —5*621, or 11*222 grammes. It is advisable to arrange some clamps and supports to hold the tube conveniently while it is cooling in the beaker of water. Instead of using water and cooling the thermometer to its temperature, we may use ice and cool it down to a tem- perature of o® C. If we do this we must, as soon as the tube is taken out of the ice, place it inside a small beaker of which we know the weight, for the temperature will at once begin to rise and some of the glycerine will be driven out Thus we should lose some of the liquid before we could complete the weighing. 262 Practiced Physics, (Ch. IX. § 37 Our next operation is to find the weight of liquid which the tube will hold at 100° C. To do this we place it in a beaker of boiling water, setting at the same time 1 receptacle to catch the glycerine which is forced out When the water has been boiling freely for some time take out the tube, let it cool, and then weigh it Subtracting the weight of the glass, let the weight of the glycerine be 10765 grammes. Thus 10765 grammes of glycerine at rcxj** C. apparently occupy the same volume — that of the thermo- meter — as 1 1*222 grammes did at 15** C. The apparent expansion for an increase of temperature of 85° (from 15**— 100°) is therefore •0425. The mean apparent expansion per i® C. throughout that range iS| therefore, '0425/85, or •0005a* This is only the coefficient of expansion relatively to glass, for the glass bulb expands and occupies a greater volume at loo** C. than at 15** C To find the true coefficient of expansion we must re- member that the apparent coefficient is the true coefficient diminished by that of the glass — had the glass at 100® been of the same volume as at 15® more glycerine would have been expelled. The coefficient of expansion of glass may be taken as -000026. Thus the true coefficient of expansion of the glycerine is '0005 26. To obtain the temperature when we take the tube firom the bath of boiling water, we may use a thermometer, or, remembering that water boils at 100° C. for a barometric pressure of 760 mm. of mercury, while an increasing pressure of 26*8 mm. of mercury raises the boiling point by i* C, we may deduce the temperature of the boiling water from a knowledge of the barometric pressure. It is better, if possible, to raise the temperature of the weight thermometer to the boiling point by immersing it in * A very convenient form of weight thermometer for accurate measurement consists of a small flask with drawn-out neck provided with a tubular collar ground to fit the neck. See Shaw, Practical Work at Cav. Lad., p. 13. Ch. IX. § 37.] Thermometry and Expansion. 263 the steam rising from boiling water, as in the hypsometer. A suitable arrangement is not difficult to make if the labora- tory can furnish ahypsometer somewhat wider than the usual ones, with a good wide opening in the top of the cover. (2) To obtain the coefficient of expansion of a piece of metal — iron, for example^ relatively to glycerine, we take a bar of the metal whose volume is obtained from a know- ledge of its weight and specific gravity, and place it in the tube before the neck is drawn out. The bar should be bent so as only to touch the tube at a few points, otherwise it will be impossible to fill the tul)e wiih the glycerine. The tube is filled after having been weighed when empty, and the weight of glycerine in it at a known tem- perature is determined. Let the temperature be o* C It is then raised to say 100** C and the weight of the glycerine within again determined. The difference between these two gives the weight of glycerine expelled. Let us suppose we know the specific gravity of glycerine; we can obtain the volume of the glycerine originally in the tube by dividing its weight by its density. Let us call this v^ We can also find the volume of the glycerine ex- pelled ; let this be z?, and let v, be the volume of the iron, at the lower temperature, v, the volume of the thermometer, /, the change in temperature, <i, the coefficient of expansion of the glycerine, j8, the coefficient of expansion of the metal, y, the coefficient of expansion of the glass. Then v=Vi+Va. When the temperature has risen /** the volume of gly- cerine is Vi(i -fa/) and that of the metal is v,(i +j8/) ; thus the whole volume of glycerine and iron will beVi(i-|-a/)-|- v,(i -h j8 /). The volume of the glass is v(i -hy /). The difference between these must clearly give the voliune of glycerine which has escaped, or v. Thus Vi(H-o/)-|-V2(n-j8/)-v(i+y^=sz?, But v=Vi+Va. Thus V|(a-y)/-|-Vj03-y)/=Z?. 264 Practical Pkysia. £Ch. tx. § 37- v,(a— y)/is the volume of glycerine which would have been expelled if the volume of the tube had been v, ; that is to say, if the tube had been such as to be filled entirely with the glycerine which was contained in it at the first weighing. This can be calculated firom the knowledge of the weight and specific gravity of the glycerine and of the value of the coefficient of expansion of the glycerine relatively to the glass. Subtract this fix>m the volume actually exi>elled. The difference is the increase in volume of the metal rela- tively to glass for the rise in temperature in question. Divide the result by the volume of the metal and the rise in tem- perature; we get the coefficient of relative expansion of the metal. Thus, let the original weight of glycerine be 1 1*222 gms., then the amount which would be expelled, due to the rise of temperature of the glycerine only, will be -457 gramme, since the coefficient of expansion of glycerine relative to glass is -0005. Suppose that we find that -513 gramme b expelled. The difference, -056 gramme, is due to the ex- pansion of the metaL Taking the specific gravity of glycerine as 1-30, the volume of this would be -043 c-c. Suppose that the original volume of the metal was 5 c.c. and the rise of temperature loo- C, the coefficient of expansion is given by dividing -043 by 500, and is, therefore, -000086. Experiments,--p^ttrmiTift the coefficient of expansion of the ffivcn liquid and of cubical expansion of the given solid. Enter results thus :— s *^ aunu. Weight of empty tube ... rw;<rm. Weight oftubefuU at 15^.5 \ \ " ^^^^ » w ft ioo**-6 Weight of liquid at i c-c Weight expelled . . Coefficient of expansion relative to glass *» ** «-". » of glass True coeffiaent of expansion Similarly for the second experiment 11-58 „ ^•52 „ •26 „ -000488 -000026 -000514 Cb. IX. $ 38.] Thermotnetry and Expansion. 265 38. The Constant Volume Air Thermometer. Determina- tion of the Coefficient of Increase of Pressure per degree of Temperature of a Oas at constant Volume. The air is contained in a closed flask or bulb, which can be heated to any required temperature. From this a tube, after being bent twice at right angles, passes vertically down- wards to a reservoir of mercury, into one end of which a plunger is fitted. A second and longer vertical tube is also screwed into this reservoir. On the tube connecting the bulb with the reservoir is a mark, which should be as near the bulb as it can conveniently be. By means of the plunger the level of the mercury in this tube is adjusted until it coincides with the mark, the bulb being kept at o^ C. by immersion in melting ice. The mercury at the same time moves in the other tube, and the difference of level of the two columns is measured by means of the kathetometer or of scales placed behind the tubes. Let this difference be 5*62 cm., and, suppose the height of the barometer to be 75*38 cm., then the pressure on the enclosed gas is that due to a column of mercury 81 cm. in height. It is of the greatest importance that the air in the bulb should be free from moisture. The bulb must, therefore, have been thoroughly dried and filled with dry air by the use of the three-way cock, drying tubes, and air-pump, as already described, (§ 16). In Jolly's air-thermometer the three-way cock is permanently attached to the tube which connects the bulb with the reservoir. The bulb is next immersed in a vessel of water which is made to boil, or, better still, in the steam from boiling water. The mercury is thus forced down the tube con- nected with the bulb, but by means of the plunger it is forced back until it is level again with the mark. At the same time it rises considerably in the other tube. When die water boils and the conditions have become steady, the 266 Practical Physics. [Ch. DC. § 38. difTerence of level in the two tubes is again noted. Suppose we find it to be 34*92 cm., and that the barometer has re- mained unchanged. The air is now under a pressure due. to 1 10-3 cm. of mercury, its volume remaining the same. The increase of pressure, therefore, is that due to 29*3 cm., and the coefficient of increase per degree centigrade is ^ ^^ ^ — , or '00362. 81 X 100 In this case it is important that the lower temperature should be o** C, for to determine the coefficient we have to divide by the pressure at 0° C, and the diflference between this and the pressure at the temperature of the room, say 15^, is too great to be neglected, as in the case of a solid or liquid. If greater accuracy be required, allowance must be made for the ctxpansion of the glass envelope, and for that portion of the air in the connecting tube which is not at the tem- perature of the bath. The same apparatus can be used to determine the coeffi- cient of increase of volume at constant pressure per degree of temperature. In this case make the first observation as before, noting at the same time the height at which the mercury stands in the marked tube. Now heat the bulb. The air will expand and drive the mercury down the one tube and up the other, thus increasing at the same time the volume of the air and the pressure to which it is subject. By with- drawing the plunger the mercury is allowed to sink in both tubes. It must, however, sink faster in the one open to the external air, and after a time a condition wiU be reached in which the difference between the levels in the two is the same as it was originally. The air in the bulb is under the same pressure as previously, but its temperature has been raised to 100® C. and its volume altered. Observe the level of the mercury in the tube connected with the bulb. If Ch. IX. § 38.] Thermometry ana Expansion. 267 the bore of this tube be known, the change of level will give the increase of volume ; hence, knowing the original volume, the coefficient of expansion per degree of tempe- rature can be found. Owing to the large amount of expansion produced in a gas by a rise of temperature of 100^ C, a tube of large bore is required. The method, however, as here described will not lead to very accurate results, for it is almost impossible to insure that the air in the bulb and that in the tube should be all at the same high temperature. In the fu^t method, on the other hand, the portion of tube occupied by air can be made very small, so as easily to be jacketed along with the bulb and kept at an uniform high temperature. The method is open to the objection that the air in contact with the mercury, and therefore the mercury itself^ is at a different temperature in the two parts of the experi- ment. The density of the mercury, therefore, is different and the increment of pressure is not strictly proportional to the difference of level. This error will be but small We have described the experiment as if air was the gas experimented with. Any other gas which does not attack the mercury may be used. JE':t}^}^in^if/.— Determine for the given gas the coefficient of the increase of pressure per degree of temperature at constant volume. Enter results thus :— T«mperatoM DUTerenMof lenl of gas of mercury o** C. 5-62 cm. 100** C 34-92 cm. Barometer . . •75*38 cm. Temperature co-efficient of pressure . . . . •00362 268 Practical Physics. [Ch. DC { l. L. The ConstaEt-pressTire Air Thermometer. Setermiiuu tion of the Coefficient of Increase of Volume per degree of temperature of a Gas at constant pressure. The measurement described in the latter part of the last section can be more accurately made in the following manner : — A glass bulb some 5 to 8 cm. in diameter opens into a short glass tube, which ends in a fine point The bulb is weighed. Suppose the weight to be w granunes. It is then filled with dry air, as in the last section, and placed in a hypsometer (§33), with its open end projecting through the cork at the top. The water in the hypsometer is heated, and after a time, when the bulb and air it contains have reached the temperature of the steam, the point is sealed oflf. If great accuracy is aimed at the bulb should, while the heating is in process, be connected with drying-tubes through a piece of indiarubber tubing. This will prevent the ingress of moisture during the heating. The temperature of the steam will be known if the height of the barometer during the experiment be read. Let it be t{*. We have thus obtained a mass of air which at a tem- perature of /i** and at a pressure given by the barometric reading fills the bulb. Now cool down the bulb, and immerse it in some liquid of known density. When under the surface of the liquid break oflf the point of the tube, carefully preserving the broken fragments of glass. Since the bulb has cooled down the pressure inside has been reduced, and the atmospheric pressure forces the liquid inside. The air in the bulb contracts. Adjust the bulb so that the surface of the liquid inside is level with that of the liquid in the vessel, and leave it for a time to take the temperature of this hquid. Let this be t^. The pressure inside the bulb is that due to the enclosed Ch. IX. § u] Thermometry and Expansion, 269 air together with the vapour pressure of the liquid used at t^^ and the sum of the two is equal to the atmospheric pressiue. If, then, the Vapour pressure of the liquid be appreciable, the pressure due to the air inside is not the same as at the time of sealing. For this f^^ason the liquid used is generally mercury, which has a vi^ small vapour pressure. We may, however, employ water without serious error, and correct the result for the vapour pressure of the water. One method of doing this is as follows : — Note the vapour pressure of water at t^ ; let it be equal to d centi- metres of water pressure. If t^ be 15®, ^ will be about 17 cm. Then depress the bulb in the water, keeping the point down, until the level of the water in the bulb is d cm. below that outside ; in this position the pressure in the bulb exceeds that of the external air by that due to a column of water d cm. in height But the pressure of the water vapour in the bulb is that due to d cm. of water ; thus the pressure of the air in the bulb is the atmospheric pressure. Thus the air in the bulb at the volume it occupies in this position and at a temperature of t^ will expand when heated to t{* so as to fill the bulb. To determine the volume of air in the bulb, close the open end of the tube with the finger or with wax and lifl the bulb out of the water; dry the outside of the bulb and weigh again, taking care to include the small fragments broken off. Let the weight be W| grammes. Then fill the bulb completely with water by placing it under water and sucking the air out through a capillary tube, and weigh again. Let the weight be w^ grammes. Then Wq— w gives the mass of water which fills the bulb, or, taking the density of water as unity, the number of cubic centimetres in the bulb ; while w,— w gives the number of cubic centimetres of water which were in the bulb when taken from the water- bath. The difference Wj— Wj is there- fore the number of cubic centimetres of air which were in the bulb at a temperature of t^ when taken out of the water- bath ; and this volume of air expands at constant 270 Practical Physics. [Ch. nL§ pressure to Wj— w at /i''. If, then, a be the coefficient of expansion at constant pressure, and Vq the volume of the same mass of air at o^ C, we have W-— w I — = Vo = Wo — w And from this equation we can find ou On reducing we have The calculation is a good deal simplified if /^ is zero^ for then Wj— w, is the volume of air at o** C, which expands, on being heated to /i**, to w^— w. Thus w,— w=(w,— Wi) This may be attained by using water cooled down to zero as the liquid in which the bulb is immersed, and this course has the additional advantage that the correction for vapour pressure is thereby greatly reduced, the vapour pressure of water at o® being '46 cm. of mercury, or about 6 '4 cm. of water, and the error committed by entirely neglecting the correction will be only about ^^. Experiment — Determine coefiicient of expansion of air at constant pressure. Enter results thus : — Height of barometer Temperature of steam, /, • Weight of empty bulb, w Temperature of water-bath, /, Weijg^ht of partly filled bulb, Wj Weight of bulb when full, w. Coefficient of expansion 7546 m. 99-8^ 16-54 3406 93*22 •00371 Ch. X.) Catorimetry. 27 1 CHAPTER X. CALORIMETRY. By Calorimetry we mean the measurement of quantities of heat There are three different units of heat which are em- ployed to express the results : (i) the amount of heat re- quired to raise the temperature of unit mass of water from o**C to i*C. ; (2) the amount of heat required to melt unit mass of ice ; (3) the amount of heat required to convert unit mass of water at 100'' into steam at the same temperature. Experiments will be detailed below (§ 39) by which the last two units may be expressed in terms of the first, which is generally regarded as the norrhal standard. Calori- metric measurements are deduced generally from one of the following observations : (i) the range of temperature through which a known quantity of water is raised, (2) the quantity of ice melted, (3) the quantity of watef evaporated or condensed ; or from combinations of these. The results obtained from the first observation are usually expre^tsed in terms of the normal unit on the assumption that the quantity of heat required to raise a quantity of water through one degree is the same, whatever be the position of the degree in the thermometric scale. This assumption is very nearly justified by experiment. As a matter of fact, the quantity of heat required to raise unit mass of water from 99** C. to 100** C. is said to be i*oi6 nomal units. The results of the second and third observations men- ticned above give the quantities of heat directly in terms of the second and third units respectively, and may therefore be expressed in terms of normal units when the relations between the various units have once been established. 272 Practical Physics. [Ch. X. § 39- 39. The Method of Mixture. Specific Heat In this method a known mass of the material of which the specific heat is required is heated to a known tempera- ture, and then immersed in a known mass of water also at a known temperature. A delicate thermometer is immersed in the water, and the rise of temperature produced by the hot body is thereby noted. The quantity of heat required to produce a rise of temperature of i^ in the cal<^meter itself, with the stirrer and thermometer, is ascertained by a preliminary experiment We can now find an expression for the quantity of heat which has been given up by the hot body, and this expression will involve the specific heat of the body. This heat has raised the temperature of a known mass of water, together with the calorimeter, stirrer, and thermometer, through a known number of degrees, and another expression for its value can therefore be found, which will involve only known quantities. Equating these two expressions for the same quantity of heat, we can deter- mine the specific heat of the material. Let if be the mass of the hot body, t its temperature, and c its specific heat; let m be the mass of the water, / its temperature initially, and Q be the common temperature of the water and body after the latter has been immersed and the temperature become steady; let m^ be the quantity of heat required to raise the temperature of the calorimeter, stirrer, and thermo- meter i^ This is numerically the same as the * water equi- valent' of the calorimeter. We shall explain shortly how to determine it experimentally. The specific heat of a substance is the ratio of the quan- tity of heat required to raise the temperature of a given mass of the substance i"" to the quantity of heat required to raise the temperature of an equal mass of water i®. If we adopt as the unit of heat the quantity of heat required to raise the temperature of i gramme of water 1% then it Ch. X. § 39.] Calorinutry. 273 follows that the specific heat of a substance is numerically equal to the number of units of heat required to raise the temperature of i gramme of that substance through i**. The mass m is cooled from t"* to &". The quantity of heat evolved by this is therefore assuming that the specific heat is the same throughout the range. The water in the calorimeter, the calorimeter itself the stirrer, and the thermometer are raised from f to V*\ the heat necessary for this is for mx is the heat required to raise the calorimeter, stirrer, and thermometer i^, and the unit of heat raises i gramme of water i*. But since all the heat which leaves the hot body passes into the water, calorimeter, &c, these two quantities of heat are equal. Hence mc(t-^ =»(»!+ »i,)(tf-i) . e=(^L±j?!i)^ .... (1) m(t— tf) ^ ' • • The reason for the name 'water equivalent' is now apparent, for the value found for my has to be added to the mass of water in the calorimeter. We may work the problem as if no heat were absorbed by the calorimeter if we suppose the quantity of water in it to be increased by m^ grammes. The quantity nty is really the ' capacity for heat ' of the calori- meter, stirrer, and thermometer. We proceed to describe the apparatus, and give the practical details of the experiments. The body to be experimented on should have consider- able surface for its mass ; thus, a piece of wire, or of thin sheet, rolled into a lump if a convenient form. Weigh it, T 274 Practical Physics. [Ch. X. S 39- and suspend it by means of a fine thread in the heater. This consists of a cylinder; a (fig. 20), of sheet copper, closed at both Ficaa ends, but with an open tube, b, running down through the mid- dle. Two small tubes pass through the outer casing of the cylinder; one is connected with the boiler, and through this steam can be sent ; the other communicates with a condenser to remove the waste steam. The cylinder can turn round a vertical axis, d, which is secured to a horizontal board, and the board closes the bottom end of the central tube. A circular hole is cut in the board, and by turning the cylinder round the axis the end of the tube can be brought over this hole. The upper end of the tube is dosed with a cork, which is pierced with two holes ; through the one a thermometer, p, is fixed, and through the other passes the string which holds the mass M. The thermometer bulb should be placed as close as possible to m. The steam from the boiler is now allowed to flow through the outer casing, raising the temperature of the mass M ; the cylinder is placed in such a position that the Ch. X. 1 39.] Calorimetry. 275 lower end of the tube in which m hangs is covered by the board. The temperature in the enclosed space will rise gradually, and it will be some time before it becomes steady, After some considerable interval it will be found that the thermometer reading does not alter, the mercury remaining stationary somewhere near 100®. Note the reading ; this is the value of t in the above equation (i). While waiting for the body to become heated the opera- tion of finding the water equivalent of the calorimeter may be proceeded witK The calorimeter consists of a copper vessel, e, which is hung by silk threads inside a larger copper vessel, f. The outside of the small vessel and the inside of the large one should be pohshed, to reduce the loss of heat by radiation. This larger vessel is placed inside a wooden box, c, to the bottom of which slides are fixed. These slides run in grooves in the wooden baseboard of the apparatus, and the box can be pushed easily under the board to which the heater is attached, being just small enough to slide under it When the box is thus pushed into position the calorimeter is under the hole in the board which has already been men- tioned ; and if the cylinder be turned so that its inner tube may come over thfa hole, the heated body can be dropped directly into the calorimeter, l is a sliding screen, which serves, to protect the calorimeter from the direct radiation of the heater, and which must be raised when it is required to push the calorimeter under the heater. A brass rod, h, is attached to the back of the box g, and carries a clip in which a delicate thermometer, k, is fixed. The thermometer bulb is in the calorimeter, a horizontal section of which is a circle with a small square attached to it ; the thermometer is placed in the square part, and is thus protected from injury by the mass m when it is immersed, or by the stirrer. The stirrer is a perforated disc of copper, frith a vertical stem. A wooden cover with a slot in it, T 2 276 Practical Physics. [Ch. X, $ 39. through which the stirrer and thermometer pass, fits over the box G. There is a long vertical indentation in the heater a, and the upper part of the thermometer can fit into this when the box g is pushed into position under the heater. Care must be taken to adjust the clip and thermo- meter so that they will come into this indentation. In determining the water equivalent it is important that the experiment should be conducted under conditions as nearly as possible the same as those which hold when the specific heat itself is being found. Let us suppose that it has been found, either from a rough experiment or by calculation from an approximate knowledge of the specific heat of the substance, that if the calorimeter be rather more than half full of water the hot body will raise its temperature by about 4®. Then, in deter- mining the water equivalent, we must endeavour to produce a rise in temperature of about 4% starting from the same temperature as we intend to start from in the determination of the specific heat Weigh the calorimeter. Fill it rather more than half full of water, and weigh it again. Let m! be the increase in mass observed ; this will be the mass of water in the calori- meter ; let f be the temperature of the water. The experi- ment is performed by adding hot water at a known tempera- ture to this and observing the rise in temperature. If the hot water be poured in from a beaker or open vessel its temperature will fall considerably before it comes in contact with the water in the calorimeter. To avoid this there is provided a copper vessel with an outer jacket The inner vessel can be filled with hot water, and the jacket prevents it from cooling rapidly. A copper tube with a stopcock passes out from the bottom of the vessel, and is bent ver- tically downwards at its open end. This tube can pass through the slot in the covering of the wooden box o close down to the surface of the water in the calorimeter. A thermometer inserted in a cork in the top of the vessel Ch. X. § 39.] Calorimetry. 277 serves to read the temperature of the hot water. For the present purpose this may be about 30^ It is not advisable that it should be much higher. Turn the tap of the hot-water vessel, and let some water xun into a beaker or other vessel ; this brings the tube and tap to the same temperature as the water that will be used. Turn the tap off, and place the calorimeter, which should be in the wooden box, with the thermometer and stirrer in position, underneath the tube, and then turn the tap again, and allow the hot water to run into the calorimeter rather •lowly. The temperature of the water in the calorimeter rises. When it has gone up about 3'' stop the hot water from flowing. Stir the water in the calorimeter well ; the temperature will continue to rise, probably about i** more; note the highest point which the mercury in the thermo- meter attains. Let the temperature be ff^ Note the tem- perature of the hot water just before and just after it has been allowed to flow into the calorimeter ; the two will differ very little ; let the mean be t'. This may be taken as the temperature of the hot water. Weigh the calorimeter again ; let the increase in mass be m' grammes. This is the mass of hot water which has been allowed to flow in, and which has been cooled from t' to ff. The heat given out is It has raised the temperature of the calorimeter, stirrer, &C., and a mass fn! of water from f to $'. The heat re- quired to do this is and this must be equal to the heat given out by the hot water in cooling, m^ being, as before, the required water equivalent Hence and 278 Practical Physics. [Ch. X. § 391 In doing this part of the experiment it is important that the apparatus should be under the same conditions as when determining the specific heat The measurements should be made, as we have said, with the calorimeter in the box, and the initial and final temperatures should be as nearly as may be the same in the two experiments. The error arising from loss by radiation will be diminished if the experiment be adjusted so that the final temperature is as much above that of the room as the initial temperature was below it. Having found the water equivalent of the calorimeter we proceed to determine the specific heat of the substance. The mass of the empty calorimeter is known ; fill the calori- meter with water from one-half to two-thirds full ; weigh it, and thus determine i», the mass of the water. Replace the calorimeter in the wooden box on the slides of the appa- ratus, and take the temperature of the water two or three times to see if it has become steady ; the final reading will be the value of t Note also the temperature of the thermo- meter p ; when it is steady raise the slide l, and push the box G under the heater, turning the latter round the axis d until the tube b is over the hole in the stand. Then by loosen- ing the string which supports it drop the mass m into the calorimeter. Draw the box back into its original position, and note the temperature with the thermometer k, keeping the water well stirred all the time, but being careful not to raise the substance out of the water. When the mercury column has risen to its greatest height and is just beginning to recede read the temperature. This gives tiie value of ft the common temperature of the substance and the water. Thus all the quantities in the equation for the specific heat have been determined, and we have only to make the substitution in order to find the value. The same apparatus may be used to determine the spe- cific heat of a liquid, either by putting the liquid into a very thin vessel, suspending it in the heater, and proceeding in the same way, allowing, of course, for the heat emitted by the Ch. X. § 39.) Catortmetry. 279 vessel, or by using the liquid instead of water in the calori- meter, and taking for the mass m a substance of known specific heat Thus c would be known, and if m be the mass of the liquid, c its specific heat, we should have Mc(T-^) = iwr(tf-/) + »»i(^-/). Hence mc(t— ^ «r, t^ Of and T having the same meaning as above. Eji^fiertmen/^'—Determmt by the method of mixtiu« the spe- cific heat of the given substance, allowing for the heat absorbed by the calorimeter &c. Enter results thus : — Name and weight of solid. Copper 32*3 gms. Temp, of solid in the heater . . 99-5 C. Weight of water .... 65*4 gms. Initial temperature of water . 12x3 C Common temp. . . . . 157 C Water equivalent of calorimeter &c. 2t> Specific Heat - x>92. Latent Heat of Water. Dbfinition. — The number of units of heat required to convert one gramme of ice at o*" C into water, without alter- ing its temperature, is called the latent heat of water. A weighed quantity of water at a known temperature is contained in the calorimeter. Some pieces of ice are then dropped in and the fall of temperature noted. When the ice is all melted the water is weighed again, and the increase gives the mass of ice put in. From these data, knowing the water equivalent of the calorimeter, we can calculate the latent heat of the water. The ice must be in rather small pieces, so as to allow it to melt quickly. It must also be as dry as possible. We may attain this by breaking the ice into fragments and putting it piece by piece into the calorimeter, brushing off 28o Practical Physics, [Ch. X. § j^ from each piece as it is put in all traces of moisture with a brush or piece of flannel The ice may be lifted by means of a pair of cmcible tongs with their points wrapped in flannel. These should have been left in the ice for some little time previously, to acquire the temperature of o® C. Another method is to put the ice into a small basket of fine copper gauze and leave it to drain for a few moments, while the ice is stirred about with a glass rod, previously cooled down to o"* C. by being placed in ice. The basket is put into the calorimeter with the ice. The water equivalent of the basket must be allowed for, being determined from its mass and specific heat Care must be taken not to put so much ice into the water that it cannot all be melted. The formula from which the latent heat is found is obtained as follows : Let m be the mass of water initially, r its temperature ; let m be the mass of ice put in, which is given by the increase in mass of the calorimeter and con- tents during the experiment ; let be the temperature when all the ice is melted, m^ the water equivalent of the calori- meter, and L the latent heat Then the heat given out by the water, calorimeter, etc, in cooling from r to is (M-|-»f,)(r-^. This has melted a mass m of ice at o® C, and raised the temperature of the water formed from o** to tf®. The heat required for this is m The temperature of the water used should be raised above that of the room before introducing the ice, and noted just before the ice is immersed. It it well to take a quantity C«. X. § 39.] Calorimetry. 281 of ice such that the temperature of the water at the end of the experiment may be as much below that of the room as it was abore it initially. We may calculate this approxi- mately, taking the latent heat of ice as 80. Thus suppose we have 45 grammes of water at 20®, and that the temperature of the room is 15®. Then the water is to be cooled down to 10% or through 10®. Thus the heat absorbed from water will be 450 units. Let us suppose we have x grammes of ice. This is melted, and the heat absorbed thereby is 80 x x. It is also raised in temperature from o^ to 10'', and the heat absorbed is XX la •*• 80 jk:+ 10 X s= 450. 90 Thus we should require about 5 grammes of ice. (If in practice we did not know the latent heat of the substance experimented upon at all, we should for this purpose determine it approximately, then use our approxi- mate result to determine the right quantity of the substance to employ in the more accurate experiment.) Experinunt, — Determine the latent heat of ice. Enter results thus : — Quantity of water • • . 47 gms. Temp, water . • • . 20° Mass of ice .... 5 gms. Common temp . . . .10^ Water equivalent of calorimeter 3*5 Latent heat of water, 79. Latent Heat of Steam, Definition. — The heat required to convert a gramme of water at 100^ C. into steam without altering its temperature IS called the latent heat of steam at loo*' C. Steam from a boiler is passed in to a weighed quantity of water at a known temperature for a short time, and the 282 Practical Physics. [Ch. X. 5 39^ rise of temperature noted. The contents of the calorimeter are again weighed, and the increase in the weight of water gives the steam which has passed in. From these data we can calculate the latent heat of the steam by means of a formula resembling that of the last section. Let M be the mass of water in the calorimeter, m^ the water equivalent, r the temperature initially, the common temperature after a mass m of steam has been passed in, i the latent heat of steam. The amount of heat given out by the steam in condens- ing to water, which is then cooled from 100' to tf®, is 'Ltn-\'m (100— tf). The heat required to raise the calorimeter with the water from r to is and these two quantities of heat are equal Hence m In practice various precautions are necessary. The steam coming directly from the boiler carries with it a large quantity of water, and moreover, in its passage through the various tubes some steam is condensed. Thus water would enter the calorimeter with the steam, and produce considerable error in the result This is avoided by sur- rounding all the tubes with jackets and drying the steam. To dry the steam a closed cylindrical vessel is employed, with two tubes entering it at the top and bottom, and a hole at the top, which can be closed by a cork carrying a thermometer. Inside this is a spiral of thin copper tubing ; the spiral emerges at the top where a glass nozzle is attached by india-rubber tubing, and terminates at the bottom in a stop-cock. The continuation of the stop- cock and the tube at the top of the cylinder are attached by india-rubber tubing to the Ch. X. 1 39 ] Calorinutty. 283 boiler ; the tube at the bottom is connected with a condenser. Thus, on putting the top of the cylinder into connection with the boiler, a current of steam passes through the copper cylinder, raising it and the spiral inside to the temperature of Ioo^ If now we put the lower end of the spiral into communi- cation with the boiler, the steam passes through the spiral, emerging through the nozzle. The spiral being kept hot at 100% the steam inside it b freed from moisture and emerges from the nozzle in a dry state. The nozzle is connected with the spiral by means of a short piece of india-rubber tubing. This should be sur- rounded with cotton wool; the cylindrical heater is placed inside a wooden box, and surrounded with wool, or felt, or some other non-conducting substance. Sometimes it is more convenient to use the boiler itself to dry the steam ; in this case the copper spiral is placed in- side the boiler, from which one. end emerges. The other end of the spiral inside the boiler is open above the level of the water. The steam, before emerging from the boiler, has to circulate through the spiral, and this dries it thoroughly. The calorimeter may conveniently take the form of a flask, or pear-shaped vessel, of thin copper, supported by silk threads inside another copper vessel Its water equivalent must be determined in the same way as has been described in the section on specific heat (p. 276). In doing this, how- ever, it must be remembered that the steam will probably raise the water to a temperature considerably higher than is the case in the determination of the specific heat of a metal. In like manner the temperature of the hot water used in finding the water equivalent should be considerably higher than that which was found most suitable in the previous experiments ; it may with advantage be some 60^ to 70*^. Now water at this high temperature may cool considerably in bang poured into the calorimeter, and care must be used to prevent loss of heat from this as far as possible. 284 Practical Physics. [Ch. X. 4 39^ In allowing the steam to pass into the calorimeter the following method may be adopted: See that the steam passes freely from the nozzle, and nott the temperature of the water in the calorimeter ; pinch the india-rubber tube connecting the nozzle with the calorimeter for an instant, and immerse one end of the nozzle under the water, then allow the steam to flow imtil the temperature has risen about 20*". Raise the nozzle until its end is just above the level of the water in the calorimeter ; again pinch the india-rubber tubing, stopping the flow of steam, and re- move the calorimeter ; note the highest point to which the temperature rises ; this will be the value of 0, the common temperature. By pinching the tube as described above, the steam is prevented from blowing over the outer surface of the calori- meter. I^ on the other hand, the tube be pinched and the flow stopped while the nozzle is under the water, the steam in the nozzle at the moment will be condensed, and the at- mospheric pressure will drive some water up into the nozzle, and this will produce error. If the calorimeter is small there is some danger that the steam from the nozzle may flow directly on to the thermometer, and thus raise its tempera- ture more than that of the surrounding water. Hiis may be avoided by the use of a calorimeter of suflScient size; Another method of avoiding this error, and one which will lead to more accurate results, is the following, which has, however, the disadvantage of requiring more elaborate apparatus. The calorimeter contains a spiral tube of thin copper, ending in a closed vessel of the same material This is completely surrounded by water, and the dry steam is passed through it instead of into the water. The water in the calori- meter is kept well stirred, and the heat given out by the steam in condensing is transmitted through the copper spiral and vessel to the water. The rise of temperature is noted as before, and when the temperature reaches its highest point, Ch. X. § 4a] Calorimetry. 285 that is taken as the common temperature of the water, spiral, and calorimeter. The heat absorbed by the spiral and vessel b determined with the water equivalent ; the quantity of water in the spiral at the end gives the mass of steam con- densed. (See Regnault's paper on the 'Latent Heat oi SteaoL' Mhnoirts de P Academic^ T. xxi.) The calculation is proceeded with in the usual way. Experiment. — Determine the latent heat of steam. Enter the results as below : — Weight of water in calorimeter • • • . 22i*3gms. Temp. • • • . • • • • 14 *5 ^' Weight of steam let in 10-4 gm& Temp, of steam given by thermometer in heater loo** Common temp, of mixture 41^ C Water equivalent of caL . • • • . 10*9 Latent beat of steam 5327 40. The Method of Cooling. To determine the Specific Heat of a Liquid. A known weight of the liquid is put into a copper vessel with a thermometer. This is hung by means of silk threads, like the calorimeter, inside another copper vessel which is closed by a lid with a cork in it supporting the thermometer. The exterior vessel is kept in a large bath of water at a known temperature, the bath being kept well stirred. It is intended to be maintained at the temperature of the room throughout the experiment ; the bath is simply to ensiure this. A small stirrer should pass through the cork which holds the thermometer, to keep the liquid well stirred. The outer surface of the inner vessel and the inner surface of the outer should be coated with lampblack. The liquid is heated up to, say, yo"" or 80'', and then put into the calorimeter. AUow the liquid to cool, and note the intervals taken by it to cool, through, say, each successive degree. If the 286 Practical Physics. [Ch. X. § 4a rate of cooling is too rapid to allow this to be done, note the intervals for each ^^ or lo"*, and calculate from these observations the mean rate of cooling for the range ex- perimented on, say from 70® to 30'. Suppose we find that, on the average, it cools 3^ in a minute. Then, if the liquid weigh 25 grammes and its specific heat be r, the quantity of heat which leaves it in one minute is 25x3 xr. Now empty the liquid out from the calorimeter and per- form a similar experiment with water instead. The water should fill the calorimeter to the same level, and be raised to the same temperature as the liquid previously used. Let us now suppose that there are 32 grammes of water, and that the temperature of the water falls through *9 of a degree in one minute ; thus the quantity of heat which escapes from the water per minute is 32 x '9 units. The quantity of heat radiated from one surface at a given temperature to another at a constant lower temperature de- pends solely on the nature and material of the surfaces and the temperature of the warmer surface.' In the two experiments described above, the surfaces are of the same nature ; thus the rate at which heat escapes must be the same for the two experiments at the same tem- peratures, ••. 2SX3xr=32X-9, ^•384. We can get the result required from the observations more quickly thus :- • Observe the time it takes the temperature to fall, say, from 60^ to 55^ in the two cases ; let it be /i minutes and /ji minutes respectively. Then the fall of temperature per minute in the two cases respectively b 5//, and s/Zj. The amount of heat which is transferred in the first case ' See Garnett, Heat, ch. u. Deschanel, Natural Phihmpin. p. 399. &c '^ Ch. X. § 4CX] Calorimetry, 287 18 scvi^ftx^ and in the second it is 5M|//|, M], m^ being the masses of the liquid and the water respectively. Thus and The effect of the vessel has hitherto been entirely neglected. Let ^ be its specific heat and m its mass, then in the first case the heat lost is in the second it is S(^»i+Ma)//^ Thus M| /) Ml i/ji ) Instead of calculating the quantity km^ we may find by ex- periment the water equivalent of the vessel and thermometer and use it instead of km. Experiment. — Determine the specific heat of the given liquid. Enter results thus : — Weight of calonin leter • • 15-13 gms. Weight of water 1094 „ Weight of liquid • • • 13-20 „ Raogw of Time of cooling of Tempenuure Liquid Water Specific heat nncorrected 70-65 115 sees. 130 sees. 733 65-60 125 „ 140 „ 734 60-55 150 n 170 „ 733 55-50 107 „ 190 „ 736 Mean specific heat (uncorrected for calorimeter) » 734 Correction for calorimeter . . -013 Specific heat of liquid •• 721 2S8 Practical Physics. [Cr. X. fif. Fig. xxiL M. Method of Cooling. OrapMo Method of Calenlatioa We may also determine the rate of cooling of a body by a graphical construction in the following manner- — Observe the temperature of the body at equal intervals of time, say every 30", and then plot a curve, taking the time for abscissa and the temperatures for ordinates. The curve will take the form of that given in fig. xxii. Let p M, p'li' be ordinates at two times represented by o M and o m' ; draw p'r parallel to 01c Then in the interval M m' the temperature £sdls by p R ; the average rate of change of tem- perature during that in- terval is P r/r p*. When the time is sufficiently small, p p' coincides with PT, the tangent to the curve at p, and the ratio p r/r p' becomes the tangent of the angle p t o ; denote it by 0. Thus the rate of cooling at any temperature can be obtained from the curve, being the tan- gent of the angle which the tangent to the curve makes with the time line. We may use the method to determine the radia- tion between two lamp-black surfaces, one of which is kept at a constant temperature while the other coob down. In any such experiment we must recollect there is very great loss of heat by convection, which we cannot avoid, so that the numbers obtained are not a true measure of the radia- tion. We take as the two surfaces those of the calorimeter, already described, and its enclosing vessel. The latter being in a large vessel of water remains constant in temperature. The calorimeter nmy take the form of a narrow rectangular vessel having considerable surface for its volume. Let the surface be measured, and let it be a sq. cm. Place a weighed quantity of water in the calorimeter, and let m be Ch. X. § M.] Calorimetry. 289 the mass of water together with the water equivalent of the calorimeter. The calorimeter should have a closely fitting cover, with two holes for the s^er and thermometer respectively, and the outer case should also be covered. Determine the temperature at equal intervals of time, keep- ing the water well stirred, and by plotting the results find the rate of fall of temperature. In drawing the curve it may be more convenient to change the scale, and to represent n seconds by one horizontal division and m degrees Centi* grade by one vertical division. In that case am tan ^ = ~- X Rate of fall of temperature ; /. Rate of fall of temperature = - tan ^ Thus the heat lost per second by the water and calorimeter in cooling is m x — tan 0, water grm degrees per second. fi And if R is the excess of the radiation per unit area emitted by the hot calorimeter over that received from the enclosure, R . A = M — tan ^ , n ^ M tn •% R = — tan ^. We may also find the radiation-difference for a differ- ence of temperature of i** by dividing r by the excess of the temperature of the calorimeter over that of the enclosure, and thus test Newton's law of cooling. Experiment, — Plot a curve of cooling for the given calori- meter, and determine from your results the radiation per unit area between the surfaces at various temperatures. Enter results thus : — Temperature of outer bath . • 15** Area of calorimeter, A . . . . 130*3 sq. cm. Mass of water + water equivalent, M . 86*8 gm. tn » .1 If. a • 34 R n •0276 tan ^. u 290 Practical Physics, [Ch. X. S m Tem BflSp oratnreof DLflTerencAof Value tfB oiimeter Tan^ K Temperatim p«i 1^ 90** •94 •02596 75* xxx>346 80^ •65 •01794 65" 276 7o« •54 •01 49 1 55* 271 60° •44 •01215 45* 270 50*^ •33 •00912 35" 260 N. Determination of the Meohanioal Equivalent of Heat^ The apparatus (fig. xxiii) consists of a strong castings supporting a vertical spindle which works in bearings, and Fig. xxiiL pa which can be driven by a large hand-wheel. There is a driving-pulley a on the axle, and near it two small pulleys b for guiding the driving-cord to the hand-wheel The cord must pass over the top of the lower wheel to the bottom of the hand-wheel, and under the higher wheel to the top of the hand-wheel. Above the driving-pulley is fixed a screw, which gears into a cog-wheel having 100 teeth, so that the cog-wheel advances one tooth for each revolution of the axle. An index is fixed so that we con tell when 100 revolutions have been completed. At the top of the spindle is fixed a cast- iron cup c, lined with cork. The cup is shown in section * See E. H. Griffiths' Thermal Measurement of Energy^ for more detaU. Ch. X. § N.] Calorimetry. 291 in the figure. Into the hollow in the cork there fits tightly a thin brass vessel in the shape of a hollow truncated cone, and within this again fits another brass vessel of a similar shape. The last vessel is provided with two pegs, which fit into a horizontal wooden wheel d, so that when the wheel is turned the vessel is turned also. A string is wound round the edge of the wooden wheel, passes over a smooth pulley, and then supports a weight p. If the apparatus is left to itself, this weight p will fall and turn the wheel and inner cup round. But if, by means of the hand-wheel, we cause the spindle to rotate in the opposite direction, it will be possible, by turning at the right speed, to keep the weight p supported so that it does not fall. The two brass cups now rub one against the other, and heat is produced. We must now calculate the work spent on fiiction in each revo- lution of the spindle when the weight p is just supported Let r cm. be the radius of the wheel, p grammes the mass of the weight, ^=981 cm. per sec. per sec=acceleration of gravity. When the weight is supported the tension of the string is vg dynes. The work spent on friction is the same whether the outer cup is in motion and the inner one at rest, or whether the inner cup is in motion and the outer one at rest In this case the work done each revolurion would be p^ x 2 tt r ergs, since 2 x- r would be the distance through which p would have to faU in order to turn the wooden wheel through one revolution. If the spindle makes n revolutions, the whole work spent on fiiction is w = «.p^.2irr ergs. Now let m be the mass of the two brass vessels, M the mass of water placed inside the iimer vessel, c the specific heat of brass. Then the brass and water together are equi- valent to M -i- rm grammes of water. If during n turns of the spindle the temperature is raised by degrees, the 292 Practical Physics. [Ch. X. § w. number of 'water-gramme-degrees' communicated to the water and brass is given by H = (m + m r). Let J be the mechanical equivalent of heat, i.e. the work in ergs that must be spent in order to produce one ' water- gramme-degree ' of heat Then in producing h * water- gramme-degrees ' we must expend j h ergs. /. J H = w, H 6(M-f»».r)' orj=^= ^'^^'^ ^^ Practical Details. — Fill the inner vessel with water up to about I '5 cm. of the top. It will be advantageous to cool the water and vessels to a temperature of about lo^ C. lower than that of the room. Work the apparatus till the temperature has risen by about 2o° C, so that it is about lo** C. above the temperature of the room at the end of the experiment If the water be not cooled, a correction must be made as follows : — When the wheel stops, note the temperature of the water, and also note the time / during which the wheel was being turned. Determine the fall of temperature (when the apparatus is at rest) which takes place during a time t Let it be </>. Then correct for the loss by radiation and convection by writing ^ + ^ ^ instead of d in the formula. The formula would then stand n ,vg , 2nr But it is much better, if possible, to make the tempera- ture at the end as much above that of the room as it was below it at the beginnmg, for in this case no correction is necessary. If the temperature of the water is adtwe the temperature Cr. X. § N.] Cahrinutry. 293 of the room on starting the experiment, the correction for loss by radiation, &c., may be made as follows : — Let the rate of falling of temperature at the initial, and final temperatures be observed. Take the mean of these rates, and multiply this by the time the experiment has lasted This product must be used instead of ^ ^ above. Two observers are required, one to turn the hand- wheel, and the other to note the revolutions of the cog- wheel and the temperature. Make a note of the time of the beginning of the experi- ment, and also the time at which each successive 100 turns of the spindle are completed. This will be a great check on accuracy of counting. Stir the water all the time, by moving the stirrer gently up and down. Do not splash. Place two or three (not six or seven) drops of oil on the inside of the outer vessel, and place the inner vessel in it before the oil has run down to the bottom of the vessel. Hang a sensitive thermometer from a clip, so as to pass through the hole in the centre of the wooden wheel, and so as to have its bulb not quite touching the bottom of the vessel. Place weights symmetrically on the wooden wheel so as to produce enough friction to raise the weight p when the wheel is worked at a convenient speed On starting, the cones slip with much greater difficulty than when once started. The following plan is convenient : — Fasten a string to p, and attach the other end to a weight q, which rests on the floor. On starting, p will not be sufficient to keep the inner cup from revolving, and Q will come into play ; as soon as the statical friction has been overcome q will fall to the ground again, and the driving-wheel must then be so manipulated that the string p Q is always slack. Great care must be taken that the string supporting p is always a tangent to the wheel. The mass of p should be about 200 grammes. 294 Practical Physics. [Ch. XL J 41 CHAPTER XI PRESSURE OF VAPOUR AND HYGROlCBTRr. 41. Daltan'i Bxperimeoit on the Preuore of Mixed Osjm. To shtw that the Maximum Pressure produced ly a Vapour in a given Space depends on l/u Temperature and not »n the Presence of Air or other Vapours in that Space. The apparatus and experiment are de- scribed in Garaett's ' Heat* A, B, G, fig. ai, are three barometer tubes. A and B are to be filled with mercuiy and inverted over the cistern of mercury d i. o contains some air above the mercury. We require, first, to explain how to fill the tubes with mercury. They must first be cleaned by washing out with dilute add, and then dried by bdng repeatedly exhausted with the air-pump and filled with air that has passed through chloride of calcium tubes. This can be done by means of a three-waycock, a^ already described (j 16). Having cleaned and dried a tube, we may proceed to fill it For this purpose it is connected with a double- neclced receiver which contains enough mercury to fill the tube, the other neck of the receiver being connected with the air-pump, and the tube and receiver are exhausted bf working the air-pump. Then by raising the end of the tube to which the receiver is attached and tilting the receiver the mercury ia allowed to flow into the empty tube from die recover. We are thus able to fill the tube with mercury free from ur without its being necessary to boil the mercorr. The three tubes should be filled in this way and inverted Oh. XL § 41.] Pressure of Vapour and Hygrometry, 295 over the mercury cistern. A convenient arrangement for the latter is a hemispherical iron basin screwed on to the end of a piece of iron tubing, the lower end of the tubing being closed Connect the open end of g by means of a bent piece of small-sized glass tubing with the drying tubes, and aUow a small quantity of dry air to flow in. The amount of air introduced should be such as to cause the mercury in g to rise to about half the height that it reaches in a and b. The quantity can be regulated by pinching the india-rubber tube which connects o with the drying tubes. Adjust in a vertical position behind the three tubes a scale of millimetres, and hang up dose to them a thermo- meter. Place a telescope at some distance off, so as to read on the millimetre scale the height at which the mercury columns stand and also the thermometer. The tube g should be so placed that it can be depressed into the iron tubing below the dstem. Mark the height at which the merouy stands in o by means of a piece of gummed paper fastened to the tube. Read on the millimetre scale the heights of a, b, and g, above the levd of the mercury in the dstem. Suppose the readings ABO 765 765 5*4 Introduce, by the aid of a pipette with a bent nozzle, a little ether into b and g, putting into each tube just so much that a smaU quantity of the liquid rests above the mercury. The mercury in b will fall. The amount of fall will depend on the temperature. Let us suppose that the new reading in b is 354 mm., then the mercury has faUen through 7^5^354 ™™- > ^^^ ^^ ether exerts a pressure equivalent to that of 41 1 mm. of mercury. The mercury in g will fall also, but not by so much as that in b;. for the pressure in g is the pressure of the ether 296 Practical Physics, [Ch. XI. § 41. vapour together with that of the contained air ; and as the mercury falls, the volume of the contained air increases and its pressure consequently decreases.* Now lower the tube g in the cistern until the level <^ the mercury in G just comes back again to the paper mailc. The volume of the contained air is now the same as before, therefore so also is its pressure. The depression of the mercury column in g below its original height is due theie^ fore to the pressure of the ether vapour. Now read the height of G on the scale \ it will be found to be about 113 mm. The column in o, therefore, has been depressed through 524—113 mm., or 411 mm. Thus b and g are depressed through equal amounts provided that the volume of air in G is allowed to remain the same. The assumption has been made that the temperature remains constant during the experiment. This will not be far from the truth in the laboratory, provided that the read- ings are taken from a distance so as to avoid the heating effects of the body ; if necessary, a correction must be applied for a change in temperature. Having made these measurements, depress b into the iron tube ; it will be found that the consequence is simply to increase the amount of condensed liquid above the sur- face of B without altering the height of that surface. The difference between the heights of the columns in a and B gives in millimetres of mercury the maximum pressure which can be exerted by ether vapour at the temperature of the laboratory. Experiment, — Determine the maximum pressure exerted by the vapour of ether at the temperature of the laboratory, and shew that it is independent of the presence of air. Enter results thus : — Height of mercury in A , , . , 765 mm, " The presence of the air in o retards the evaporation of the ethef a considerable time must therefore be allowed for the mercury to arrive at its final level. 354 H 411 n 524 »» "3 }) 411 if Ch. XI. § 41.] Pressure of Vapour and Hygrometry. 297 Height of mercury m B — initially ...... 765 mm after introduction of ether • • Pressure of ether vapour Height of mercury in G — initially ....•• after introduction of ether • • Pressure of ether vapour . . , Temperature i5**-5 throughout The volumenometer described in §26 will afford us another means of testing Dalton's law. Introduce a smaU quantity of water or other liquid into the bulb e (fig. 16), and screw it on. As the water evaporates the pressure will increase and the level of the mercury change. When it has become steady read the level in both tubes, and note the height of the barometer. Alter the position of the tube a and take another reading, and thus obtain a series of corresponding values of volume and pressure. Let us suppose the volume of the flask is known, so that v, the actual volume occupied by the air, can be found. Allowance must be made for the volume occupied by the water, which of course changes slightly ; this is easily done by weighing the flask empty, then with the water, at the beginning and end of the experi- ment These last two will differ, but very slightly, owing to the evaporation. From the mean of the two weights and the weight of the empty flask we can obtain the average volume of the water, which will be sufficient for our present purpose. Write down the reciprocals of the observed values of v, and then plot a curve with these reciprocals as abscissae and the observed pressures as ordinates. If Dalton's law is true, or has the same actual error at all pressures, the curve will be found to be a straight line, as a. b in fig. xxiv, cutting the axis of v in b. Let p m be any ordinate, and through o draw o Q parallel to a b, cutting p m in q. Let o B =/oj then 298 Practical Physics. [Ch. XI. S 41. OM = l/v, P M=/, Q M =/ -/o- Now, from the figure, — = tan Q o M = ^, say, k being some constant ; O M /. {p—po) v = constant. Thus if we diminish the observed pressure by a constant quantity /oi t^^ product of the difference and the volume is constant. The observed pressure / is therefore the sum of a con- Fic xxiv. stant pressure /o and a pressure/,, which satisfies Boyle's law — i.e. the actual pressure is that due to the air obeying Boyle's law together with a constant pressure, that of the aqueous vapour saturating the space at the given tempera- ture. On varying the temperature the same law will be found to hold, but the pressure /© will be different for different temperatures ; and if Dalton's law is true, the values of /o for different temperatures will correspond exactly with those given in Regnault's table of saturation-pressures dt aqueous vapour. Ch. XL § 41] Pressure of Vap<mr and Hygrometry, 299 • In carrying out the experiment it is very important that the temperature should be constant, as the pressure of the vapour changes greatly with temperature. Time must in each case be given for the air to become saturated. Experiment — ^Verify Dalton's law. HYGROMETRT. Pressure of Aqueous Vapour} — ^The determination of the amount of water contained in the atmosphere as vapour is a problem of great importance, especially to meteorology. There are several ways in which we may attempt to make the determination, and the result of the experiment may also be variously expressed. The quantity of water which can be contained in air at a given temperature is limited by the condition that the pressure ^ of the vapour (considered independently of the pressure of the atmosphere containing it) cannot exceed a certain amount, which is definite for a definite temperature, and which for temperatures usually occurring, viz. between ~ 10° C and -f-3o** C, lies between 2 nmi. of mercury and 31*5 mnL Dalton concluded, from experiments of his own, that this maximum pressure, which water vapour could exert when in the atmosphere, was the same as that which the vapour could exert if the air were removed, and indeed that the dry air and the vapour pressed the sides of the vessel containing them with a pressure entirely independent one of the other, the sum of the two being the resultant pressure of the damp air (see the pre fious experiment, § 41). This law of Dalton's has been shewn by Regnault to be true, within small limits of error, at different temperatures for saturated air, that is, for air which contains as much vapour as possible ; and it is now * In the first edition of this work the words * pressure' and 'tension' were used, in accordance with custom, as sjmonymous. In this edition it is intended to use the term * pressure ' only in referring to aqueous vapour. 300 Practical Physics. [Ch. XL § 41. a generally accepted principle, not only for the vapour of water and air, but for all gases and vapours which do not act chemically upon one another, and accordingly one of the most usual methods of expressing the state of the air with respect to the moisture it contains is to quote the pressure exerted by the moisture at the time of the ob- servation. Let this be denoted by e ; then by saying that the pressure of aqueous vapour in the atmosphere is e^ we mean that if we enclose a quantity of the air without altering its pressure, we shall reduce its pressure by ^, if we remove from it, by any means, the whole of its water with- out altering its volume. The quantity we have denoted by e is often called the pressure of aqueous vapour in the air. Relative Humidity, — From what has gone before, it will be understood that when the temperature of the air is known we can find by means of a table of pressures of water vapour in vacuo the maximum pressure which water vapour can exert in the atmosphere. This may be called the saturation pressure for that temperature. Let the tem- perature be / and the saturation pressure e^ then if the actual pressure at the time be e, the so-called fraction of saturation will be — and the percentage of saturation will be ?^?-?, This is known as the relative humidity. Dew Point, — If we suppose a mass of moist air to be enclosed in a perfectly flexible envelope, which prevents its mixing with the surrounding air but exerts no additional pres- sure upon it, and suppose this enclosed air to be gradually di- minished in temperature, a little consideration will shew that if both the dry air and vapour are subject to the same laws of contraction from diminution of temperature under con- stant pressure,* the dry air and vapour will contract by the same fraction of their volume, but the pressure of each will be " The condition here stated has been proved by the experiments ol Regnault, Herwig, and others, to be very nearly fulfilled in the case ol water vapour. Ch. XI. § 4*1 Pressure of Vapour and Hygronutry. 301 always the same as it was originally, the sum of the two being always equal to the atmospheric pressure on the outside of the envelope. I( then, the pressure of aqueous vapour in the original air was e^ we shall by continual cooling arrive at a tempe- rature — ^let us call it r — at which t is the saturation pressure ; and if we cool the air below that we must get some of the moisture deposited as a cloud or as dew. This temperature is therefore known as the dew point If we then determine the dew point to be r, we can find e^ the pressure of aqueous vapour in the air at the time, by looking out in the table of pressures e^ the saturation pressure at r, and we have by the foregoing reasoning 42. The Chemical Method of Determining the Density of Aqueous Vapour in the Air. It is not easy to arrange experiments to determine directly, with sufficient accuracy, the diminution in pressure of a mass of air when all mobture shall have been ab- stracted without alteration of volume, but we may attack the problem indirectly. Let us suppose that we determine the weight in grammes of the moisture which is contained in a cubic metre of the air as we find it at the temperature / and with a barometric pressure h. Then this weight is properly called the actual density of the aqueous vapour in the air at the time, in grammes per cubic metre. Let this be denoted by d^ and let us denote by 8 the specific gravity of the aqueous vapour referred to air at the same pressiu-e e and the same temperature /, and moreover let w be the density of air at o® C and 760 mm. pressure expressed in grammes per cubic metre. Then the density of air at the pressiu-e e and temperature /, also ex- Dressed in grammes per cubic metre, is equal to f^ — . " 760(1 -f-o/) 302 Practical Physics. [Ch. XI. § 42. where a = coefficient of expansion of gases per d^^ee centigrade, and therefore . Zew 760(1 + a/) 760(1 + 0/) . or ^ = i — ^— f a, w Now w is known to be 1293 and a = '00366 ; .^^760(1+00366/)^^ .... (I) 1293 S If, therefore, we know the value of 8 for the conditions of the air under experiment, we can calculate the tension of the vapour when we know its actual density. Now, ios water vapour which is not near its point of saturation ^ is equal to '622 for all temperatures and pressures. It would be always constant and equal to '622 if the vapour followed the gaseous laws lip to saturation pressure. That is however, not strictly the case, and yet Regnault has shewn by a series of experiments on saturated air that the for- mula e = 760(1 + -00366 1)^ suffices to give accurately the 1293 X '622 pressure when d is known, even for air which is saturated, or nearly so, with vapour. We have still to shew how to determine d. This can be done if we cause, by means of an aspirator, a known volume of air to pass over some substance which will entirely absorb from the air the moisture and nothing else, and determine the increase of weight thus produced. Such a substance is sulphuric acid with a specific gravity of i*84. To facilitate the absorption, the sulphuric acid is allowed to soak into small fragments of pumice contained in a U-tube. The pumice should be first broken into fragments about the size of a pea, then treated with sulphuric acid and heated to redness, to decompose any chlorides, &c., which may be contained in it. The U-tubes may then be filled with the fragments, and the strong sulphuric add poured on till the Ch. XI. 5 4a.] Prtssurt of Vapour and Hygrometry. 303 pumice is saturated; but there must not be so much acid that the air, in passing through, has to bubble, as this would entail a. finite diSerence of pressure on the two sides before the air could pass. Phosphoric anhydride may be used instead of sulphuric acif}, but in that case the tubes must be kept horizontal. Chloride of calcium is not sufficiently trustworthy to be used in these experimmts as a complete absorbent of moisture. The arrangement of the apparatus, the whole of which can be put together in any laboratory, will be understood by the fig. 32. As aspirator we may use any large bottle, A, having besides a thermometer, two tubes passing airtight through its cork and down to the bottom of the bottlb. One of these tubes is bent as a syphon and allows the warn to run out, the flow being regulated by the pinch- cock T ; the other tube is for the air to enter the aspirator ; its opening being at the bottom of the vessel, tiie flow of air is maintained constant and independent of the level of the water in the bottle. The vessel b, filled with fragments of freshly fused chloride of calcium, is provided with two tubes through an 304 Practical Physics, [Ch. XI. § 4x airtight cork, one, connected with the aspirator, passing just through, and the other, connected with the drying tube d^ to the bottom of the vessel. This serves as a valve to prevent any moisture reaching the tubes from the aspira- tor. The most convenient way of connecting up drying tubes is by means of mercury cups, consisting of short glass tubes with a cork bottom perforated for a narrow tube ', over this passes one limb of an inverted U-tube, the other limb of which is secured to one limb of the drying tube either by an india-rubber washer with paraffin or, still bettor by being thickened and ground as a stopper. A glance at the figure will shew the arrangement The drying tubes can then be removed and replaced with facility, and a perfectly airtight connection is ensured. The space in Uie little cups, M, M, M, M, between the narrow tubes and the limbs of the inverted U's is closed by mercury. Care must be taken to close the ends of the inverted ITs with small bungs during weighing, and to see that no globules of mer- cury are adhering to the glass. The connecting tubes c between the drymg tubes should be of glass and as short as possible. Two drying tubes must be used, and weighed separately before and after the experiment; the first will, when in good order, entirely absorb the moisture, but if the air is passed with too great rapidity, or if the acid has become too dilute by continued use, the second tube wiU make the fact apparent A thermometer, x, to determine the tem- perature of the air passing into the tubes is also necessary. To take an observation, the tubes are weighed and placed in position, the vessel a filled with water, the syphon tube filled, and the tube at the end of the drying tubes closed by means of a pinch-tap. Then, on opening the tap at t, no water should flow out ; if any does there is some leak in the apparatus which must be made tight before proceeding further. When assured that any air supplied to the aspirator will pass through the drying tubes, the observation may be begun. The water is run out slowly Ch. XI. § 42.] Pressure of Vapour and Hygrometry, 305 (at about the rate of i litre in ten minutes) into a litre flask, and when the latter is filled up to the scratch on the neck it is removed and weighed, its place being taken \yj another flask, which can go on filling during the weighing of the first This is repeated until the aspirator is empty, when, the weight of the empty flasks being ascertained, the total weight of water thus replaced by air can be found. The height h of the barometer must be determined at the beginning and end of the experiment During the observa- tion the thermometer x must be read every ten minutes, and the mean of the readings taken as the temperature / of the entering air ; the thermometer in the aspirator must be read at the end of the experiment ; let the reading be /. If the aspirator a is but small, it can be refilled and the ex- periment repeated, and we may of course determine, once for all, the volume of water which can be run out of the aspirator when filled up to a certain mark in the manner thus described; but as an exercise it is better to re-determine it for each experiment From the weight of water run out, with the assistance of Table 32 (Lupton, p. 28) we can determine the volume v of air taking the place of the water in the aspirator, v being measured in cubic metres. This air is evidently saturated with water at the temperature /; its pressure is the baro- metric pressure, and therefore the pressure of the dry air in it is H— tf/r, tf being the saturation pressure at /. When it entered the drying tubes this air had a pressure h»^, and its temperature was /, e being the pressure whose value we are seeking. The volume of the air was, therefore, Hence, if w be the increase of weight of the drying tubes in grammes, we shall have for d the actual density of the moisture in the air ; a^^ "m M , t • • • ( j|) H— gf i-f-g/ ^ ^^' H— * ' i-l-a/ 3o6 Practical Physics. [Ch. xi. § 4s. We thus obtain the quantity d; substituting its value from equation (i) above, we get 1 293 X '62 2 _ (h — ^)(i+a/)w 760(1 +a/) ^ "" (H^^)(l+a/)v' or e ^ 760 i4-a/^ w H— tf 1293 X '622 * H—Cf ' V (4) Experiment.^'DeitTmxnt, the density of the aqueous vapour in the air, and also its pressure. Enter results thus : — Temperature of air 2i°7 Temperature of aspirator Volume of aspirator \ Gain of weight of tube (i) « (2) 99 Total #«i6'o8. . 56061 oc . '5655 gm. • xxDii gm. '5666 gm. 43. Dines*s Hygrometer. Wet and Dry Bulb Thermometers. Dines's Hygrometer is an instrument for directly deter- mining the dew-point, i.e. the temperature at which the air in the neighbourhood of the instrument is completely satu- rated with aqueous vapour. It consists of a thermometer placed horizontally, so that its stem is visible while its bulb is enclosed in a box of thin copper through which cold watet can be passed from a reservoir attached to the instrument by turning the tap at the back. The tap is full on when the side marked o is upward, and shut off when that marked s is upward. The bulb of the thermometer is placed dose to the top of the box which encloses it, and the top of the box is formed of a plate of blackened glass, ground veiy thin indeed, in order, as far as possible, to avoid any difference of temperature between the upper and undo Ch. XL § 43-1 Pressure ^ Vapour and Hygrotnetry. 307 sur&ces, and so to ensure that the temperature of the thermometer shall be the same as that of the upper sur&ce of the glass. The temperature of the box is cooled very gradually by allowing water, previously cooled by adding ice, to pass very slowly from the reservoir along the tube. As soon as the surface of the glass is at a temperature below that of the dew point, a deposit of dew can be observed on it This can be easily noticed by placing the instrument so that the glass surface reflects the light of the sky, and accordingly presents a uniform appearance which is at once disturbed by a deposit of dew. The temperature /, say, at which this occurs is of course below the dew-point The film of moisture is then allowed to evaporate, and when all has disappeared the temperature is again read— let it be (, This must be accordingly above the dew-point Now allow the water to flow only drop by drop, cooling the surface very slowly indeed, and observe the same phenomena again, until / and / are not more than one or two tenths of a d^;ree apart Then we know that the dew-point lies between them, and by taking the mean of the two obtain an accuracy sufficient for practical purposes. The fall of temperature can in some cases be made so slow that a fugitive deposit forms and disappears at the same temperature, in which case the temperature of the dew-point is indicated by the thermometer as accurately as the variation of the quantity to be observed permits. It is important that the observer should be as for as possible from the glass surfoce during the observation, in order to avoid a premature deposit of moisture. To this end a telescope must be mounted so as to read the thermo- meter at a distance, placing a mirror to reflect the scale of the thermometer to the telescope. We may thus determine the dew-point, but the usual object of a hygrometnc observation is to determine the pres sure of aqueous vapour in the air at the time of observing. X a 308 Practiced Physics. [Ch. XI. $ 43. We may suppose the air in the neighbourhood of the de- positing sur^u:e to be reduced to such a state that it wffl deposit moisture, by altering its temperature merely, without altering its pressure, and accordingly without altering the pressure of aqueous vapour contained in it We haye, therefore, only to look out in a table the saturation pressure of aqueous vapour at the temperature of the dew-point and we obtain at once the quantity desired, viz. the pressure of vapour in the air before it was cooled. We may compare the result thus obtained with that given by the wet and dry bulb thermometers. In this case the observation consists simply in reading the temperature of the air /, and the temperature Z' of a thermometer whose bulb is covered with muslin, which is kept constantly moist by means of a wick leading from a supply of water. The wick and muslin must have been previously boiled in a dilute solution of an alkali and well washed before being mounted, as otherwise they rapidly lose the power of keep- ing up a supply of moisture from the vessel. The pressure t!' of aqueous vapour can be deduced from the observations of / and f by Renault's formula^ (available when / is higher than the freezing point) •'=^- ooo9739/'(/-/')--S94i(/-0 —•ooo8(/— 0(^—755) where tf is the saturation pressure of aqueous vapour at the temperature t^ and b is the barometric height in millimetres. Kxperiments, — Determine the dew-point and the pressure of aqueous vapour by Dines's Hygrometer, and also by the wet and dry bulb thermometer. ' The reduction of obseryations with the wet and dry bulb ther- mometers is generaUy effected by means of tables, a set of which is issued by the Meteorological Office. The formula here quoted is Kegnault's formula (Ann. de Qiimie, 1845) as modified by Jdin^ See Lupton, table 35. Ch. XI. § 43.1 Pressure of Vapour and Hygrotnetry. 309 Elnter the results thus : — Appearance of dew . • • • 47^*1 F. Disappearance of dew . • . • 47°75 Dew-point 47*'*42 Pressure of aqueous vapour deduced . 8*33 mm. Pressure of aqueous vapour from wet and dry bulb • • • . • 8*9 mm 44. Begnaiilt*8 Hygrometer. Rq;naiilt's hygrometer consists of a brightly polished thimble of very thin silver, fonning the continuation of a short glass tube to which the silver thimble is attached by plaster of paris or some other cement not acted upon by ether. Through a oork fitting tightly into the top of the l^ass tube pass two narrow tubes of glass, one (a) going to the bottom of the thimble, the other (b) opening at the top of the vessel just below Uie code; also a sensitive thermo- meter 80 placed that when the cork is in position, the bulb (which should be a smaU one) is close to the bottom of the thimble. I^ then, ether be poured into the thimble until it more than covers the thermometer bulb, air can be made to bubble through the liquid either by blowing into the tube (a) or sucking air through (b) by means of an aspirating pump of any sort The passage of the air through the ether causes it to evaporate and the temperature of the liquid to foil in consequence, while the bubbling ensiu-es the mixing of the different layers of liquid, and therefore very approxi- mately, at any rate, a uniform temperature of silver, ether, and thermometer. The passage of air is continued until a deposit of dew is seen on the silver, which shews that the tem- perature of the silver is below the dew-point The thermo- meter is then read, and the temperature of the apparatus allowed to rise until the deposit of moisture has completely disappeared, when the thermometer is again read. The temperature is now above that of the dew^int, and the 3IO Practical Physics, \ctL. XL § 44- mean of the two readings so obtained may be taken as the temperature of the dew-point, provided that there is no more difference than two or three tenths of a degree centi- grade between theoL In case the difference between the temperatures of ap- pearance and disappearance is a laige one, the method of proceeding suggested by Regnault should be adopted. The first observation will probably have given the temperature of dew appearance within a degree; say the observation was 5®; pass air again through the ether and watch the ther- mometer, and stop when a temperature of 6° is shewn. Then aspirate slowly, watching the thermometer all the dme. Stop as each fifth of a degree is passed to ascertam if there be a deposit of dew. As soon as such a deposit is formed, stop aspirating, and the deposit will probably dis- appear before the temperature has risen 0^*3, and we thus obtain the dew-point correct to 0^*1. The thermometer should be read by means of a tele- scope some 6 feet away from the instrument, and every care should be taken to prevent the presence of the observer producing a direct effect upon the apparatus. It is sometimes very difficult, and never very easy, to be certain whether or not there is a deposit of dew on the silver, the difficulty varying with different states of the light It is generally best to have a uniform light-grey background of paper or cloth, but no very definite rule can be given, practice being the only satisfoctory guide in the matter. A modification of R^^nault's apparatus by M. Alluard, in which the silver thimble is replaced by a rectangular brass box, one fiice of which is surrounded by a brass plate, is a more convenient instrument ; the contrast between the two polished surfiu:es, one of which may be covered with the dew while the other does not vary, enables the appear- ance of the deposit to be judged with greater facility. The method of using the instrument is the same as for Renault's. The dew-point being ascertained as described, the Ch. XI. ^ 44-1 Pressure of Vapour and Hygrometry, 311 pressure of aqueous vapour corresponding to the tempera- ture of the dew-point is given in the table of pressures based on Regnault's experiments,' since at the dew point the air is saturated with vapour. We have ab-eady seen (p. 301) that we may take the saturation pressure of vapoiu: at the dew- point as representing the actual pressure of aqueous vapour at the time of the exi>eriment. Experiment. — Determine the dew-point by Regnault's Hy- grometer, and deduce the pressure of aqueous vapour. Calcu- late also the density of air in the laboratory at the time of observation. Enter results thus : — Appearance of dew .... 47*** i F. Disappearance ... • 47 75 Dew-point 47 '42 Pressure of aqueous vapour . . • 8 '33 mm. CHAPTER Xtl. PHOTOMETRY. Thb first experiments to be performed in optics will be on the comparison of the intensities of two sources of light We shall describe two simple methods for this, Bunsen's and Rumford's, both founded on the law that the intensity of the illumination from a given point varies directly as the cosine of the angle of incidence upon the illuminated surface and inversely as the square of the distance of the siu^e from the luminous point So that if I, I' be the illuminat- ing powers of two sources distant r, r' respectively from a given surface, on which the light from each fiedls at the same angle, the illumination from the two will be respectively I/f* and V\f'^^ and if these are equal we have I : r=f* : f'\ so that by measuring the distances r and r we can find the ratio of I to r. * Lnptoo'i Tables^ No. 34, 312 Practical Physics. [Ch. XII. § 44. Now this supposes that it is possible to make the illumi- nation from each source of light the same by varying the distances of the two sources from the screen. As a mattei of iasXy this is not necessarily the case ; in performing the experiment we compare the two illuminations by the effect produced on the eye, and that effect depends partly on the quantity of energy in the beam of light reaching the eye, partly on the nature of the rays of which that beam is composed. To define the intensity of a beam, we require to know, not merely the quantity of light in it, but also how that light is distributed among the differently coloured rays of which the beam is composed. Any given source emits rays, probably of an infinite number of different colours. The effect produced on the eye depends on the proportion in which Uiese different colours are mixed. Lf they are mixed in different proportions in the two beams we are considering, it will be impossible for the effect of each of the two, in illuminating a given surface, ever to appear the same to the eye. This constitutes the great difficulty of all simple photo- metric measurements. Two different sources of light, a gas flame and a candle for example, emit differently coloured rays in different proportions ; the gas light contains more blue than the candle for the same total quantity of light, and so of the two spaces on which the illumination is to be the same, the one will appear bluish, the other reddish. Strictly, then, two different sources of light can only be compared by the use of a spectro-photometer, an instrument which forms the light from each source into a spectrum and then enables the observer to compare the intensity of the two for the different parts of the spectrum. One such in- strument will be described in a subsequent section (§ 67). 45. Bonsen's Photometer. Two standard sperm candles (su p. 23) are used as the standard of comparison. These are suspended from the ano Ch. XII. $ 45] Photometry. 313 of a balance and counterpoised so that the amount of wax burned can be determined at any moment without moving the candles. This arrangement is also useful in keeping the flames nearly in the same position, for as the candles bum down the ann supporting them rises. The balance is to be placed so that the candle-flames aie vertically over the zero of the scale of a photometer bench in a dark room. As a source to be compared with these, we use a gas-flame, the supply of gas being regulated and measured thus : — The gas is passed from a gas-holder, where the pressure can be altered by altering the weights on the cover, through a meter, u, flg. 33, which measures the quanti^ of gas passed through. One complete revolution of the needle corresponds to ^th of a cubic foot of gas, so that the numbers on the dial pa^ed over in one minute give the number of cubic feet of gas which pass through the meter in an hour. The gas enters at the middle of the back of the meter and leaves it at the bottom, passing thence to a governor, g, which consists of an inverted bell, partly sunk in water and counterpoised so that the conical plug attached to its top is very dose to the conical opening of the entrance pipe q. Any increase of pressure of the gas in the bell raises the bell, narrows the aperture, and diminishes the supply until the pressure falls again. By this means the pressure of the gas at the burner is maintained constant 314 Practical Physics. Ch. XIL § 45. The exit pii>e from the bell passes to a tube with two stopcocks s, s'. The stopcock s' is provided with a screw adjustment for regulating the supply of gas with extreme nicety; the stopcock s can then be used, being always either turned on full or quite shut, so as to always reproduce the same flame without the trouble of finely adjusting eveiy tima Between these two stopcocks is a manometer ic for measuring the pressure of the gas as it bums. In stating, therefore, the gas-flame employed, we have to put down (i) the burner employed ; (2) the pressure of the gas ; (3) the amount of gas passing through the meter per hour.' The gas passes from the stopcocks to the burner, which is fixed on one of the sliding stands of the photometer bar, so that the plane of the flame corresponds to the fiducial mark on the stand. On another sliding stand between the burner and the candles b placed the photometer disc, which consists of a grease spot upon white paper. The method consists in sliding the photometer disc along the scale until the spot appears of the same brightness as the rest of the paper ; the intensities of the lights are then proportional to the squares of their distances fix)m the disc The observations should be made by viewing the disc fix)m either side, as it will often be found that when the spot and the rest of the disc appear to be of the same brightness when viewed from one side, they will difler con- siderably when viewed from the other. This is due, in part, at any rate, to want of uniformity in the two sur&ces of the paper of which the disc is made ; if the diflerence be very marked, that disc must be rejected and another used. In all cases, however, observations should be made fix>m each side and the mean taken. The sources of light should be screened by blackened ' In order to test the < lighting power of gas * with a standard argand burner, the flow through the meter must be adjusted to 5 cabic feet per hoar by means of the micrometer tap. Ch. XIL § 45.1 Photonutry. 315 screens^ and the position of the disc determined by several independent observations, and the mean taken. The lights most be very nearly of the same colour, otherwise it will be impossible to obtain the appearance of equality of illnmination over the whole disc. (This may be tried by interposing a coloured glass between one of the lights and the disc) Instead of trying to find a position in which the disc presents a uniform appearance on one side, the position in which it appears the same as viewed from two corresponding points, one on each side, may be sought for. For additional details see the ' Gas Analysts' Manual,' p. 40, §§ 61, 84. Experiment.'^om^zxt the illuminating power of the gas- flame with that of the standard candle. Additional experiments. — (a) Compare the intensities of the candles and standard axgand burner — (i) Du-ectly. (2) With a thin plate of glass interposed between one source and the disc This wiU give the amount of light lost by reflection and by the absorption of the glass. By rotating the glass plate the variations in the loss at different angles may be tested. (3) With a thin plate of glass between one source and the disc, and a thick plate on the other side. This wiU enable you to deteimine the amount of light lost by the absorption of a thickness of glass equal to the difierence ol the thicknesses of the two plates. ifi) Obtain two burners and arrange them in connection with a three-way tube. Cover one up by a screen, and measure the intensity ojf the other. Then interchange them, and so obtain the intensity of each separately. Then place them together so that the two flames unite, and measure the intensity of the combined flame and its relation to the sum of the intensities of eadi. {c) Test the intensity of the light from the same amount off gas used in difierent burners. Enter results thus : — Gas burning at the rate of 5 cubic feet per boor. Candles « h 16*3 gms. « 3i6 i Fracttcal Phystcs. [Ch. XIL § 45^ lUuidiiCaiioeof Mom distanc* of Ratio of WmniMring gu 75 31 5-85 68 29 5-49 60 25 576 52 11 5-59 46 19 5*86 Mean ratio of flluminating powers 571. 46. Bmnford's Photometer. The apparatus for making the comparison consists simply of a bar, at the end of which a ground g^ass or paper screen is fixed, and on which a support is made to slide, carrying the gas jet or other source of light On the bar, and in front of the screen, is placed a wooden rod, about 3 inches from the screen. The two lights to be compared are placed one on the sliding support and the other on the table at a fixed distance (taking care that both are the same height), the positions being so adjusted that the two shadows of tfie rod thrown on the screen are just in contact with each other without overlapping. The screen must be turned so that it makes equal angles with the direction of the light from each source. The distance of the sliding light has to be adjusted so that the two shadows are of the same depth Consider a unit of area, e.g. a square centimetre, of each shadow A and B ; let the distance of the unit of area of a from the two soiurces of light be ^, x, and let the distance of the unit of area of the shadow b from the same sources be y^ Y respectively. Then the unit of area of a is illuminated only by the one soiurce of light, distant x from it, and therefore its illumination is I/x*, where I is the illumination per unit area at unit distance from the source. The unit of area of B is illuminated only by the source of light at dii- tancex and the illumination therefore b r/j^» when F is the illumination per umt area at unit distance from the second source^ CH.xa§46.] Photometry, 317 . Hence, since the flluminations of the shadowed portions of the screen are equal, rTy^ •• p-y If the two unit areas considered be immediately ad- jacent to the line of junction of the shadows, then we may measure x and y from the same point. Hence the ratio of the intensities of the two sources is the square of the ratio of the distances of the two soiurces from the line of contact of the shadows. The method has the advantage that the observations do not need a dark room. The shadows may be so arranged that the line of contact is on the middle line of the bar on which the one source slides, and accordingly the distance may be measured along the bar. The other distance may be measured by a tape. The arrangements necessary for determining the rate at which the gas is being burnt or the quantity of wax con- sumed are described in section 45. Experiment — Compare the illuminating power of the gas- flame and standard candle. Enter results thus : — Candle bums at the rate of 8*1 gms. per hour. Gas M » 5 cubic feet per hour. Distance of gM Distance of candle Ratio of illtiminatin{ powers 128-5 39*5 10-5 98 30-5 10-4 Mean ratio of illuminating powers 10*45 OF T- ^ ' V t v/ T "^t " . i' y ' $ 1 8 Practical Physics. [Ch. Xin, § 47 CHAPTER XIII. REFLEXION AND REFRACTION— MIRRORS AND LENSES. Nearly all the methods used in optical measurements are indirect The quantity required is deduced by calculation from the quantities actually measured, or the law to be demonstrated is inferred from the actual observations by a process of reasoning. This is illustrated by the following experiment on the law of reflexion and by the experiments on focal lengths. The law of refraction may also be verified by the measurements of the refractive index of a transparent medium. 47. Verifloation of fhe Law of Beflexion of Light In order to prove the law, that the angle which a reflected ray makes with the normal to a plane surface is equal to the angle made by the incident ray with the normal, and that the two rays are in the same plane with the normal, two methods may be adopted : — (i) The direct method, in which the angles of incidence and reflexion are measured and compared, and the positions of the rays determined. (2) An indirect method, in which some result is verified which may be theoretically deduced on the assumption that the law holds. The following experiment is an example of the second method. It may be proved, by assuming the law of reflexion, that an image of a luminous point is formed by a plane mirror at a point on the normal to the plane sur&ce drawn through the luminous point, and at a distance behind the mirror equal to the distance of the luminous point from the fixmt of the mirror. This we can verify experimentally. Ch. XIIL 547.1 Mirrors and Lenses. 319 Take as the luminous point the intersection of cross-wires mounted on a ring, which can be placed in any position in a dip. We can place another similar cross in the exact position occupied by the image in the mirror of the first, in the following manner. Scrape a horizontal strip of the silvering off the back of the mirror and place the one cross in front, so that on setting the eye on a level with the cross, half of the image is seen coming just to the edge of the silvering. Then place the other cross behind, so that it can be seen through that part of the glass from which the silvering has been scraped. Place this second cross so that the upper half of it can be seen through the gap, and so that the intersection of the second appears to coincide with the image of the in- tersection of the first In order to determine whether or not this is really the case, move your eye from side to side across the first cross-wire, then if the second cross and the image are coincident, the two will appear to move together as the eye moves, and will remain coincident wherever the eye is placed. If, however, the actual cross is nearer to the mirror than the image, then on moving the eye to the right the two will appear to separate, the further, viz. the image, going to the right hand, the real cross to the left Place, then, the second cross so that on moving the eye firom side to side no separation between the cross and the image occurs. It is then in exactly the same position as that occupied by the image of the first cross in the mirror. Let the first cross be placed at a distance of i foot (about) firom the reflecting surfiice of the mirror. Measure the distance by means of a pair of compasses and a scale, and measure, also, the distance between the same surface of the mirror and the second cross, which has been accurately placed to coincide with the image of the first in the mirror. Then displace the second cross fix>m coincidence with the image and replace it and read the distance again in order 320 Practical Physics. [Ch. XIIL % 47. to ascertain the limit of accuracy to which your observatioD can be carried. Repeat three times. The experiment may be very conveniently made with a piece of unsilvered plate glass instead of the mirror. The image of the first cross formed by reflexion at the surface of the glass is generally sufficiently bright to permit of the second cross being accurately placed to coincide with it. If the glass is very thick, allowance must be made for the dis- placement of the image of the second cross as seen through the glass. A corresponding allowance may, of course, also be necessary in the case of the mirror whose thickness will alter the apparent position of the reflected image of the first cross. Two vertical pins in stands may be used instead of cross- wires, and the upper part of the second one may be viewed directly over the top of the mirror, while the lower part of the image of the first is seen in the mirror. In order to verify that the image and object are on the same normal to the mirror, place the eye so that the image and object are in the same straight line with it, and notice that the image of the eye is in the same line too, no matt^ how £ur from or how near to the mirror the eye be placed ; this can only be the case if the line is a normal. In case the result obtained does not apparently confirm the law of reflexion, the discrepancy may be due to the foct that the mirror is cylindrical or spherical and not truly plane To distinguish between the cases, repeat the experiment, mov- ing the eye vertically up and down instead of horizontally. ' Experiment.— Verdy the truth of the law of reflexion <rf light. Enter results thus : — Dbtance of object Distance of image 75 cm. 75 cm. 65 n 63 „ 8o'5 I. 78 „ 715 „ 71-5 ». 61 .1 59 r Ch. Xiri. g 47.] Mirrors and Lenses. 321 The following method of finding the angle of a prism is another illustration of the law of reflexion :— Place the prism on a sheet of paper attached to a draw- ing-board. Let BAG {fig. zyi) be its trace, a being the angle to be measured. Stick a pin (p) vertically into the board at some distance from A, in such a position that images by reflexion can be obtained from the f'"-*': * » anH * r respectively. Deter of these images as or proceed as folio image, moving you it is seen as neaily as pos- sible in a line with A ; and place pins at Q, B, so that the image of q ^ p, the edge a, and the pin at Q appear in one straight line, while the image, the edge, and the pin at r are seen in another. Then a ray p A foiling on one face very close to A is reflected along A Q, while an almost coincident ray incident on the other face is reflected along a r. Join a p, a q, a r, and measure with a protractor the angles q a r and bag. It will be found that qar = 2BAC; that is, that the angle between the reflected rays is twice the angle of the prism. This can be proved to be a consequence of the law of reflexion. Experim£nt—Tit\trtn\nt the positions of the images of a pin formed by the light reflected from the two surfaces of a prism, and thence measure the angle of the prism. 322 Practical Physics. [Ch. XIIL § 4. Fig. 04. 48. The Sextant The sextant consists of a graduated circular arc, bc (fig. 24), of about 60**, connected by two metal arms, a b, A 0, with its centre a. ad is a third movable ano, which turns round an axis passing through the centre a, at right angles to the plane of the arc, and is fitted with a damp and tangent screw. A vernier is attached to this arm at D, and by means of it the position of the arm with '•• reference to the scale can be determined. The ver- nier is generally con- structed to read to 15", A plane mirror, m, is attached to this arm and moves with it The plane of the mirror passes through the centre of the circular arc and is at right angles to the plane of the scale. The mirror is known as the index glass, and is held by adjustable screws in a frame which is rigidly connected to the arm a d. By means of the screws it can be placed so that its plane is accurately perpendicular to that of the arc At P on the arm a c is another mirror called the horizon glass, also secured by adjustable screws to the arm. Its plane should be perpendicular to that of the arc and parallel to that of the movable mirror m when the index at i) stands at the zero of the scale. The upper half of the mirror p is left unsilvered. At o on the arm a b is a small telescope, directed towards the mirror f. The axis of the telescope is parallel to the plane of the arc, and by means of a screw at the Ch, XIIL § 48.] Mirrors and Lenses. 323 back of the instrument the telescope can be moved at right angles to this plane, so as to direct its axis towards the silvered or unsilvered part of the horizon glass. This is placed in such a position that its normal bisects the angle a f G^ and hence a ray of light, parallel to the plane of the sextant, travelling along a f, is reflected by the horizon glass parallel to the axis of the telescope. Let p a be such a ray reflected by the mirror m in direction a f, and suppose p to be some distant object the position of which we wish to observe. Let the telescope be so placed with reference to the plane of the instrument that light from a second distant object q, also travelling parallel to the plane of the sextant, can enter the telescope through the unsilvered part of the glass f. Then an observer, looking tha)ugh the telescope, will see the point q directly, and the point p after reflexion at the two mirrors m and f. The telescope is fitted with cross-wires, and by altering the position of the arm a d the image of p can be made to coincide with that of Q in the centre of the field of view. Let us suppose this adjustment made. Then by re- flexion at the two mirrors the ray p a has been made to coincide in direction with the ray q f. Hence, the angle between pa and qf is twice the angle between the two mirrors. But when the index read zero the two mirrors were parallel, so that twice the angle between the two mirrors is twice the angle through which the arm and vernier have been turned from zero. In many instruments the graduations are numbered to read as double of their real value ; each degree is reckoned as two degrees and so on, so that, if the instrument be in adjustment, the reading of the vernier gives us directly the angle between p a and q f, that is, the angle which the two distant points p and Q subtend at the observer's eye. The requisite adjustments are : — (i) The plane of the index glass m should be at right angles to that of the graduated arc. V 2 324 Practical Physics. [Ch. XIII. § 48. (2) The plane of the horizon glass f should also be at right angles to that of the arc (3) The axis of the telescope should be parallel to the plane of the arc. (4) The index and horizon glasses should be parallel when the vernier reads zero. We proceed to consider how to make these adjustments. The two glasses are held in their frames by screws, and can be set in any position by altering these screws. (i) Place the eye close to the index glass and look towards the glass so as to see part of the arc c d and its reflexion, meeting at the surface of the glass. If the two, the arc and its image, appear to be in the same plane, then the glass is perpendicular to that- plane. If, however, the image appears to rise out of the plane of the arc, the upper portion of the glass leans forward towards the eye, while if the image appears to drop below the plane of the arc, the glass leans back away from the eye. Adjust the screws till the arc and its image appear to be in the same plane ; then the plane of the glass is at right angles to that plane. (2) To set the horizon glass. Hold the instrument so as to view directly with the telescope some distant point -a star if possible. On turning the index arm round, an image of the point, formed by reflexion at the two glasses, will cross the fleld. If the two glasses be accurately parallel, this image can be made to coincide exactly with the object seen by the direct rays. If the plane of the horizon glass be not at right angles to that of the arc, so that the two mirrors can never be parallel, the image will appear to pass to one side or the other of the object By altering the adjusting screws of the horizon glass, the image seen after two reflexions, and the object seen directly, can be made to coincide in position. When this is the case the two mirrors are strictly parallel, and the horizon glass, therefore, is at right angles to the plane of the arc Ch. XIIL § 48.] Mirrors and Lenses. 325 (3) To set the axis of the telescope parallel to the plane of the arc For this it is necessary that the ring to which the telescope is fixed should be capable of being moved about an axis parallel to the line of intersection of its plane with that of the arc The eye-piece of the telescope is usually fitted with two cross-wires, very approximately paraUel to the plane of the arc, and one wire at right angles to these, passing through their middle points. The line joining the centre of the object glass to the middle point of this wire is the optical axis of die telescope. Hold the instrument so as to view ^o distant points, such as two stars, the one directly and the other by reflexion at the two glasses, and incline it to the plane through the eye and the two stars in such a way that the two images seen in the telescope appear to coincide at the point in which the third wire cuts one of the two parallel wires. Then, without moving the index glass, in- cline the plane of the instrument until the image of the star seen direcdy falls on the intersection of the third wire and the other of the two parallel wires. If the image of the second star again coincides with that of the first, it follows that the optical axis of the telescope is parallel to the plane of the arc ; to make the two parallel the position of the telescope with reference to the arc must be adjusted until it is possible to observe such a coincidence. (4) To set the two mirrors parallel when the vernier-index reads zero. It will be found that one of the glasses with its frame and adjusting-screws can be moved about an axis at right angles to the plane of the arc Set the vernier to read zero and clamp it, and direct the telescope to some distant point If the two glasses are parallel this point, and its image after reflexion at the two mirrors, will appear to coincide. If they do not coincide they can be made to do so— -supposing adjustments (i) and (2) have been made — by turning the movable mirror about the axis just 326 Practical Physics. [Ch, XIII. § 48. fpoken of, and when the coincidence is effected the mirron will be parallel, while the vernier reads zero. Instead, however, of making this last adjustment, it is better to proceed as follow^ to determine the index Gnx>r ot the instrument Direct the telescope to a distant point and turn the index glass until the image of the point, after reflexion at the two mirrors, coincides with the point itself as seen directly. Clamp the vernier and read ; let the reading be a. If the instrument were in perfect adjustment, the value (^ a would be zero. Suppose, now, we find that when proceed- ing to measure the angular distance between two distant points, as already described, the scale and vernier reading is ^, then the angular distance required is ^— a. Generally it gives less trouble to determine the index error than to set the mirrors so that there is no such error. It may, of course, happen that the value of a is na- tive — in other words, that to bring a point and its image into coincidence we have to push the vernier back beyond the zero of the scale ; for this reason the scale graduations are continued beyond the zero. It is important for accurate work that the two images which are brought into coincidence should be about equally bright Now, the light Arom one has suffered two reflexions, each of which somewhat diminishes its intensity. If, then, the two distant objects are unequally bright, we should choose the duller one as that to be viewed directly. Again, we have said already that the telescope can be moved in a direction at right angles to the plane of the arc In its normal position the axis of the telescope will pass throu^ the boundary between the silvered and unsilvered parts of the horizon glass. Half the object-glass will accordingly be filled with direct light, half with reflected. If the direct light is very much stronger than the reflected, we can, by moving the telescope, still keeping its axis parallel to the plane of the circle, place it so that the reflected rays fill Ch. XIII. § 48.1 Mirrors and Lenses. 3^7 more than half and the direct rays less than half the object glass, and thus reduce the brightness of the direct and increase that of the reflected image. There are also shades of coloured glass attached to the instrument, which can be interposed in the path of either pencil and so decrease its intensity. The instrument is frequently used to observe the altitude of the sun or of a star ; and in this case the horizon, if it is visible, forms one of Uie distant points, and when the in- strument is adjusted, the image of the sun's lower limb should appear to coincide with this. If the horizon be not visible, an ' artificial horizon ' is ob- tained by reflexion from some horizontal sur&ce — that of pure mercury in a trough is most frequently used. For consider two parallel rays s a, s' b (fig. 25) coming from a distant object, and let s'b be reflected at b firom a horizontal surface cd. b a appears to come from the image of the distant object formed by reflexion at cd, and if an ob- server with a sextant at a determine the angle between the distant object and its image, he will measure the angle SAB. But since s a is parallel to s' b and the angle a bd is equal to s' b c, the angle s a b is twice the angle s' bC| that is, twice the altitude of the distant object If mercury be used for the artificial horizon, it should be covered with a piece of carefully worked plate glass. After one observation the cover should be taken up and turned round and a second taken. The mean of the two will be free from any small error which might arise from the faces of the glass not being parallel Sometimes a piece of glass, which can be carefully levelled, is used instead of the mercury 328 Practical Physics. [Ch. XIII. § 48. Experiments. (i) Test the accuracy of the various adjustments of the sex- tant. (2) Measure the angular distance between two distant points. (3) Measure the altitude of a distant point, using an arti- ficial horizon. £nter results thus : — Index emr Angubr diminct 2' 15'' y^^ 35' 30" ^' 30" 32^ 35' 15 ^' 30" 32** 35' 15 Mean 2' 25'' 32** 35' 20" True angular distance 32^ 32' 55^' Similarly for observations of altitude. O. Befraction of Light through a Plate and throngrb a Prism. The path of a pencil of light through a plate or a prism may be traced and the law of refraction verified by a graphical construction in the following manner. Place a rectangular block of glass, which should be of considerable size — say 8 or 10 cm. square by i cm. high — on a sheet of paper fastened to a drawing-board, and mark its position a b c d (fig. xxv) on the paper. Draw a line p q meeting the glass obliquely, and stick two pins vertically into the board at two points some distance apart in p q. On looking obliquely through the opposite face (c d) of the glass the two pins will be seen, and it will be usually possible to place the eye in such a position that the one may appear exactly behind the other. Do this, and stick two more pins into the board in front of the glass in such a way that these two are seen in the same strai^t line as the first two, so that all four appear to be in line one behind the other. Draw with a ruler a line R s through Ch. XIII. § o.] Mirrors and Lenses. 329 the feet of the last two, and let it meet the surface of the glass in R. Join Q R. Then a ray of light falling on the glass in the direction p Q is refracted into the glass along Q R, and on emergence travels along r s. On completing the figure it will be seen that pq is parallel- to rs. Draw mqn normal to the glass at Q. Then M Q f is the angle of incidence <f, and n q r the angle of refraction ^'. To Verify the Law of Refraction (»«, that sin ^jsin ^' is constant) and find the Refractive Index. With Q as centre and q r as radius describe a circle cutting Q p in p. Draw p m perpendicular to the normal Q M. Measure the distances p m and R N, and take the ratio. Then sin » _P M vQR_PJ* sin^' Qp RN rn' /or Q P = Q R. J30 Practical Physics. [Ch. XIIL § a Take a second incident ray p, q, incident at a different angle, and determine the refracted ray Ri S| in the same way. Then we shall have sin ^1 _ P| M, sin^i'" r;n;' and it will be found that the ratios— and ^i-^ are equal R N Ri N, Thus this ratio is a constant for all angles of incidence, and the value of this constant is the refractive index. We have thus verified the law of refraction and found ^, the index of refraction. To Illustrate the formation of a Caustic Curve by Refraction. Stick a vertical pin into the board in contact with ihe block at o. Let on be a normal meeting the opposite FiG..=vL &ce in N, and along that fece mark off a number of points p,,p„P(.. . . suchtliatNP, = p, P2 = p, PjZE. . , IS I cm. suppose. Ch. xilL § O.] Mirrors and Lensu 331 At each of the points N^f-i,?), • • • place pins in contact with the block. Look at the block from a little distance, and place another series of pins in the board successively in such positions that the pin o, each of the pins N, Pi, Pj, . . in turn, and the corresponding pins, m, Qi,Q), . . . of this next series appear successively in straight lines. Remove the block, join q^ Pj, q^ P2, Qa Pa* • • • » and produce each of these lines backwards to the point in which it meets the next preceding line. Let these points be Ri, r^i Ra* Then a ray travelling in the block along o p^ b refracted so as to emerge along p^ Q2, and so for the other rays. Again, if we can suppose two consecutive emergent rays p^Qt, P3 Qa, to reach the eye, these rays wiU appear to diverge from Rs> and the position of the image of o which the eye sees when looking along Q2 p^ will be R^. In reality, the rays P3 Q2, Pa Qa are too far apart to be treated as consecutive rays ; we should have to suppose incident rays to fall on all the points of the glass between n and p, and draw all the emergent rays. In this way we should obtain a series of points, such as Ri, . . . R«, all lying on a curve, each point being the intersection of two consecutive emergent rays. This curve is called the caustic curve, and to it all the emergent rays are tangents, while the virtual image of o seen in the direction of any given ray is the point in which that ray touches the caustic curve. I^ then, the figure be constructed as already described, and a curve drawn to touch all the emergent ra}^, this curve will be the caustic The same figure can be used to verify by a geometrical construction the law of refraction. To find the Refractive Index. The following is another method of finding ft : — Make a mark at a point a (fig. xxvii) on one face of the block. This may be done by sticking on to it a small piece of sealing-wax. Place the block on the table, and stick a pin upright into the board in such a way that Ai, 532 Practical Pkysics. tCH. XIII. § o the bead of the pin, is at the same height as a. Od look- ii^ through the block the reflected image of the pin and the image of a can both be seen. Move the block about until these two, when viewed directly from a point behind the pin, appear to coincide. In this case a b Ai, cutting the block in b, will be normal to the block, and if a' is the refracted image of a, it is also the reflected image of a,. Since the light is nearly directly incident, we know that B A = /I B a' ; and since a' is the reflected image of Ai, Hence to find >i, measure the thickness of the block and the distance ba, of the pin from the block. The ratio of the two is the refractive index. To Verify the Law of Refiexiort. Similar experiments can be performed to verify the la* of reflexion. In this case let p q (fig. xxviii) be an inddmt Ch. XIII. §0-1 Mirrors and Ltmes. 333 ray falling on the block at Q. At two points on this ray stick iwo pins vertically into the board, and then look at the reflected images of these pins. Move your eye about until these images appear ■- '*-" ~ straight line, and in two other upright pi joining Rs it will be the surface in Q, and to make an angle with the nonnal at Q equal to that made by p Q. If, more- over, a number of incident rays pq,, pQt . . . be taken, and the directions of the reflected rays de- termined, it will be found that these all meet in a point p,, and if p p, be joined, cutting the face of the block, or this face produced, in n, p P, is at right angles to that face, and is bisected in n. By replacing the block by a prism, the laws of refraction through the prism may be verified. Refraction through a Prism. Draw a ray pq (fig. xxix) incident obliquely on a prism. The direction of the refracted ray and of the ray in the prism can be found in exactly the same way as in the case of a plate. By the aid of a protractor the angles, ^ and ■^, of inci- dence and emergence respectively can be found, and the deviation d, which is the angle between the incident ray p q 334 Practical Physics, [Ch. Xlll. § a and the emergent ray r s. If the angle of the prism, i, be measured, we can verify the formula D = ^ + ij, - I. Moreover, by varying ^ from zero up to grazing incidence, for which ^ = 90°, we can examine the changes in the ,„ ..J. deviation. We shall find that as the angle of inci- dence increases the devia- tion decreases at Urst, then reaches a minimum value, and afterwards increases ^ain as the angle of inci- dence is still further in- creased up to grazing inci- dence. In the position of minimum deviation we can shew that the incident and emei^nt rays are equally inclined to the sur&ce of the prism, so that 9 = il' for this position. More over, in general we .lave ^', ^ being the angles which the ray in the prism makes with the normals to the two faces. Hence, in the position of minimum deviation, for which f = J/, we have ^' = i '■ ^ = Hd + '") ; .*. u = ^'1.* = sin Hp + 0. sin ^' sin \ i The images of the pin seen by refraction will usually be slightly coloured, but unless the dispersion of the prism is very great this will not seriously affect the results. The Ck. XIII. § o.] Mirrors and lenses. 335 instrument described in § 62, the spectrometer, enables us to make the measurements above described to a much higher degree of accuracy than is possible with the ruler and pencil. Mxferiment, — Trace the path of a ray of light through a plate of glass, and hence verify the law of refraction, and find the refiactive index of glass. Trace the caustic curve formed by rays diverging from a point and emerging from glass. Trace the path of a ray through a prism, and verify the formulae <^' + >(.'-/ <f> f >/^ * D + /. Shew that in the position of minimum deviation ^ >■ ^t and find the refractive index of the prism. On Optical Measurements, Many of the simpler optical experiments described below depend on the determination of the positions of some luminous object and its real image formed after reflexion 01 refraction. A formula is obtained expressing the quantity sought for, e.g. the focal length of a lens, in terms of distances which can be readily determined. These are measured and their values substituted in the formula ; the value of the quantity in question is determined by calculation. . Now, in almost every case, the formula is one giving the relation between the position of a point and its geo- metrical image, and to obtain this the assumption is made that we are only concerned with a small pencil, the axis of which b incident directly on the reflecting or refracting surfru:es. If this be not the case, there is no such thing as a point image of a point The rays diverging from a given point of the object do not all converge again exactly to one and the same point For each point in the object we have — supposing still that the incidence is direct— a least circle of aberration through which all the rays from that point pass, and the nearest approach to an image is the 33^ Practical Physics. [Ch. xiil. § ^ figure formed by the superposition of all these least circles of aberration, which will be a representation of the object, more or less blurred, and differing in position from the geometrical image. Now, frequently this happens with the images produced by the optical combinations with which we shall have to da The pencils which go to form the various images are not small pencils incident directly, and the phenomena are thus complicated by the eflfects of aberration. Thus, for example, we may require the radius of a con- cave mirror, three or four inches across and six or eight inches in radius ; or we may be experimenting with a lens of an inch or so in diameter and only one or two inches in focal length. In both these cases we should meet with aberration difficulties. We shall see best how to allow iox this in each separate experiment There is one measurement common to many optical experi- ments, the mode of making which may best be described herCs Two objects — the one may be a lens, the other a screen on which an image is focussed — are attached to the supports of an optical bench described below. This is graduated, and the supports possibly are fitted with verniers ; at any rate, there is a mark attached to them, the position of which, with reference to the scale of the bench, can be found. We can thus find easily the distance between the two fixed marks on the supports ; but suppose we require the dis- tance between the screen and one face of the lens. To obtain this we must know their positions with reference to the fixed marks. Now, the apparatus is generally constructed so that the central plane of the lens and the plane of the screen respectively are in the same vertical plane as the marks in question, so that, neglecting the thickness of the lens, the distance between the marks is, as a matter of fact, identical with the distance required. But for some purposes this is not sufficiendy accurate. We may, for example, wish to consider the thickness of the lens in our measurements Ch. XIII. § 48,] Mirrors and Lenses. 337 In this case» take a rod with two pointed ends, and mea- sure carefully its length. Let it be a. Put one end against the screen and move up the support carrying the other surface, until this is in contact with the other end of the rod. Let the distance between the marks on the supports, as read at the same time by the scale and vernier, be b. Then, clearly, if in any other position of the supports the distance between the marks on them is r, the distance between the surfaces is ^+a— ^, for a was the distance between them in the first position, and ^—^ is the distance by which it has been altered. We may make the same measurement by the following slightly different method which can be used conveniently for determining the distance between two objects measured parallel to any fixed scale. Fix securely to the vernier of the scale a stiff piece of wire, and bend it until its end comes in contact with one of the objects in question, and read the vernier. Now move the vernier with the wire fixed relatively to it, along the scale, until the same end of the wire comes in contact with the second object, then read the vernier again. The difference between the two readings is the distance required This will be foimd a convenient way in making the measurements, described in § 49, if the mirror can be fitted to one of the supports of the optical bench. Of course, if the distance required be only small, the simplest method of all is to use a pair of compasses and take it off along a finely divided scale. 49. Heasnrement of the Focal Length of a Concaye Krror. This may be obtained optically by means of the formula ' V u r f • For the foromla reoaifed in this and the next chapter we may lefer to Glarctrook, Physual Optics, chap. nr. Z 338 Practical Physics. [Ch. XIII. § 49. /being the focal length, and r the radius of the surface, u and V respectively the distances from the surface of an object and its image ; u and v can be measured, and thenr or/ calculated. In practice the following modification of the method will be found most convenient It depends on the fact that when the image of an object formed by a concave mirror coincides with the object itself, then the object is at the geometrical centre of the spherical surface. Place a needle in a clip and set it in front of the mirror ; place the eye some distance further away from the mirror than the needle. An inverted image of the needle will be seen, unless the needle has been placed too close to the mirror. Adjust the position of the needle relatively to the mirror, so that the point of the image coincides with the point of the needle. When this is the case the image will be of the same size as the object The adjustment can be made as finely as necessary, either by moving the eye about and noting whether the relative positions of image and needle vary, or by using a strong magnifying lens, and noticing whether botii needle and image are in focus at the same time. If the aperture of the mirror be very large, and its sur£ice not perfectly spherical, it may be impossible to see the image when using the lens, in consequence of the aberration of die rays from the outer portions of the surface. These defects may, in some cases, be corrected by covering the mirror with black paper, leaving at the centre only a smaD hole, which may be either oblong or circular. When the position of the needle has been carefully adjusted, measure its distance from the reflecting surfiEu:e by means of a pair of compasses and a scale, if the radius be small, or by the method already described if the mirrpr be fitted to the optical bench. The result gives the length of the radius of the minor furface. Half of it is the focal length. Ck. xm. § 49.] Mirrors and Lenses, 339 Experiment — Determine the radius of curvature of the given mirror, and check your result by the use of the sphero- meter. Enter results thus : — Radius of curvature by optical observations 19*52 cm. Radius of curvatiure by spherometer . 19*8 cm. 50. Measurement of the Badins of Curvature of a Eefleotmg Snrfeu^e by BeflexioiL The method of § 49 is applicable only when the rieflect- ing surface is concave, so that the reflected image is real. The following method will do for either a concave or convex surface. Fig. a6. Let o, fig. 26, be the centre of the reflecting surface, o c X the axis. Suppose two objects a', a" placed at equal distances on each side of o c x, and at the distance o x from o. Images of these two points will be formed by reflexion at points a\ a" on the axes o a', o a", such that (calling the points where the axes oa, oa' cut the spherical surface C,c") I _ I _ 2 A'c' dd or and I A^ _I __ 2 5^C^ oc' a"c" oc z 2 340 Practiced Physics. [Ch. XIII. § sa Now, the points being very distant, and therefore d a.' very nearly equal to c x, we may assume that the straight line of tf" cuts the axis o ex at a point x where I ex ex oe and for the size of the image, we have (') a a" ox ,. a'a"~ox* ••:••• l«^ Hence, if e x = A, oc = r, a' a" = l, cxzs^x^ andtf'tf"=A we get from (i) i=i-^ (3) h X r ^ Hence A r X r* • • ^ ■ f A X and A r+A L r+A* L From these two equations Ar *== X= aA+r aA+r Place a small, finely divided scale s s' immediately id front of the reflecting surface (but not so as to prevent aD the light falling upon it) i.e. place it horizontally to cover nearly half the reflecting surfisice, and observe the images Ch. XIIL § 5a] Mirrors and Lenses. 341 af^ of' and the scale s s' by means of a telescope placed so that its object-glass shall be as nearly as possible in the middle of the line joining a a' ; we may with sufficient accuracy suppose the centre of the object-glass to be at the point X. Join xa', xa'' and let the lines xa', xa" cut the scale s s' in l' and \J'^ and let / denote the length \J iJ' of the scale intercepted by them. Then we get /^jU/_ A or A + JC aA + f' / ^— X '-'' .AT 2A + r aA+r Lr or 2(A+ry aA/ L — 2/ r= The formula proved above refers to a convex surface ; if the surface be concave we can find similarly the equation 2A/ L+2/ To make use of this method to find the radius of curva- ture of a surface, place the surface opposite to, but at some distance from, a window. Then place horizontally a straight bar of wood, about half a metre in length between the surface and the window, fixing it with its ends equidistant from the sur^ice, and at such a height that its reflexion in the surface is visible to an eye placed just below the bar, and appears to cross the middle part of the surface. Fix a tdesoope under the centre of the bar, with its object-glass 342 Practical Physics. (Ch. XHL § 5a in the same vertical plane as the bar, and locus it so as to see the image reflected in the surface. It is best that the whole of the bar should be seen re- flected in the sur&ce. If this cannot be secured, two wdl- defined marks, the reflected images of which can be dearly seen, should be made on the bar. These may be obtained by fixing two strong pins into the upper edge, or by laying on it two blocks of wood with clearly defined edges. In any case the reflected image should appear in the telescope as a well-marked dark object against the bright background of the reflexion of the window. If it be more convenient to work in a dark room, arrangements must be made to illuminate the bar brightly, so that its refiexioa may appear light against a dark background. Now place against the reflecting surface a finely gra duated scale— one divided to half-millimetres or fiftieths of an inch will do — arranging it so that one edge of the image of the bar is seen against the divided edge of the scale. If the curvature of the surface be considerable, and the magnifying power of the telescope not too great, the scale will be fairly in focus at the same time as the image of the bar. At any rate, it will be possible to read the graduations of the scale which the image of the bar appears to cover. This gives us the length / of the above formula. Measure the length of the bar or the distance between the two marks — this we call l; and measure with a tape the distance between the reflectiDf surface and the centre of the object-glass of the telescope— this gives a. Then the formula gives us r. In some cases it may be possible to see more than one reflected image of the bar; e.g. if a reflecting surface be one surface of a lens, we may have a reflexion firom the back surface as well as from the front A little consideration enables' us to choose the right image. Thus, if the firsl surface is convex, the reflected image will be erect and wiD, Cb. XIII. 1 5a] Mirrors and Lenses. 343 therefore, appear inverted if we are using an astronomical telescope. Experiment — Determine the radius of the given surface, checking the result by the use of the spherometer. Enter results thus : — Sorfikoe CooviK A- 175*6 cm. L- 39-4 cm. /« 2*06 CUL r- 20'5 cm. Value foond by spherometer 20^ cm. Measurement of Focal Lengths of Lenses. The apparatus generally employed to determine the focal length of a lens is that known as the optical bench It consists simply of a horizontal scale of considerable length, mounted on a substantial wooden beam, along which upright pieces can slide^ and to these are severally attached the lens, the luminous object, and a screen on which the image formed by the lens is received. These sliding-pieces carry verniers, by which their position with reference to the scale can be determined. The position of each face of the lens relatively to the zero of the vernier is known or can be found as described on p. 337. 51. Measurement of the Foeal length of a Convex Lens. — First Method. For this purpose a long bar of wood is employed, cany- ing at one end a ground-glass screen, fixed at right angles to the length of the bar. A stand, in which the lens can conveniently be fixed with its axis parallel to the length of the bar, slides along it, and the whole apparatus is port- able, so that it can be pointed towards the sun or any other distant object Place the lens in the stand and withdraw to a dark comer of the laboratory ; point the apparatus to a distant 344 Practical Physics. [Ch. XIII. § 51. well-defined object — a vane seen through a window against the sky is a good object to choose if the sun be not visible — and slide the lens along the bar until a sharply defined image of the object is formed upon the ground glass. Since the object is very distant, the distance of the lens from the screen is practically equal to the focal length, and can be measured either with a tape or by means of graduations on the bar itself The observation should, of course, be made more than once, and the mean of the measurements taken. 52. Measurement of the Focal Length of a Conyez Lens. — Seoond Hethod. Mount on one of the stands of the bench a diaphragm with a hole in it across which two fine threads are stretched, or, if more convenient, a piece of fine wire grating, or a pin in a vertical position with its point about the centre of the hole. Place a light behind the hole, taking care that the brightest part of the light is level with the hole and exactly behind it, while the light is as close to the hole as may be. In the second stand place the lens, fixing it so that its centre is on the same level as that of the hole inlhe dia- phragm, while its axis is parallel to the length of the bench. In the third stand fix an opaque white screen ; a piece of ground glass or unglazed paper is most suitable. For the present purpose the objects can generally be fixed on their respective stands so as to occupy with sufficient accu- racy the same relative positions with regard to the zeros of the verniers, and thus the distances between the different objects in question can be obtained at once, by reading the verniers and subtracting. If the distance between the first and third stand be more than four times the focal length of the lens, the latter can be placed so that there is formed on the screen a dis- tinct image of the object in the first stand Move the stand carrying the lens till this is the case. Then measure Ch- XIII. I 52.) Mirrors and Lenses. 34 5 by means of the verniers fixed to the stands, or as de- scribed on p. 337, the distance, m, between the object and the first surface of the lens and the distance, v^ between the image and the second sur&ce. Then if we neglect the thickness of the lens the focal length/ is given by the formula^ f V u The values of v should be observed for at least three different values of u. Experiment, — Determine by the methods of this and the preceding sections the focal length of the given lens. Enter results thus : — Lens A. Approximate focal length ({ 51) 58 cm. By method of $ 52^ 105*6 128-8 58-02 994 1401 5815 85-0 181-9 57.92 Mean value of focal length 58*03 53. Measurement of the Focal Length of a Convex Lens.— Third Hethod. The methods already described for finding the focal lengths of lenses involve the measurement of distances from the lens surface^ and con- sequently a certain amount of error is caused by neg- lecting the thickness of the glass of which the lens is composed. This be- comes very important in '' the case of short-focus lenses and of lens combinations ' GUucbrook, Physical Optics^ chap. iv. 346 Practical Posies. [Ch. XIIL { 53. The following method avoids the difficulty by renderir^ the measurement from the lens surfaces unnecessary. We know that for a convex lens, if «, 9 are the distances respectively of the image and object from the principal points * of the lens e f (fig. 27), and/its focal length ; then f u'lf u and V being on opposite sides of the lens. Now, if we have two screens a b, cd a distance / apart, and we place the lens bf, so that the two screens are in conjugate posi- tions with regard to it, then u-k-v^l^ provided we neglect the distance between the two principal points. In strictness, v+t' is not equal to /, as the distances • and V are not measured from the same point, but from the two principal points respectively, and these are sepa- rated by a distance which is a fraction of the thickness of the lens. Thus, if / be the thickness of the lens, it may be shewn that the distance between the principal points is ^^^/, if we neglect terms involving /• ; the value of this for glass k about ^ The image of a cross-wire or a piece of wire-grating at the one screen a b will be formed at the other, c d. Nov we can find also another position of the lens, b' f', between the screens, such that the image of the cross- wire or grating is again focussed on the second screen. This will evidently be the case when the lens is put so that the values of u and t are interchanged. Let uf and i/ be the values which »and f assume for this new position of the lens, and let the distance v'— » or v^f/ through which the lens has been moved be a Then we have u V f * See Penaiebury't Lenses emd Systems cf Lenses, pt. 39 d te^ Ch. XIII. § 53.J Mirrors and Lenses. 347 But Hence a 2 Substituting • •/- ^ > so that the focal length may be determined by measuring the distance between the screens (which must be greater than four times the focal length), and the distance through which the lens has to be moved in order to transfer it from one position in which it forms an image of the first screen on the second, to the other similar position. This latter measurement should be made three or four times and the mean taken. For screens, in this case, we may use small pieces of wire gauze mounted in the circular apertures of two of the stands of the optical bench, or we may fix two pins with their points at the centres of these apertures. The coincidence of the image of the first object with the second may be determined by the parallax method described in §§ 47 and 49 ; or the following very convenient arrange- ment may be adopted : — In the apertures of the two stands of the optical bench mount two pieces of gauze, as suggested above, setting one of them ¥dth its wires horizontal and vertical, and the other with its wires inclined at an angle of 45*' to these directions. On the stand carrying the gauze on which the image is to be received, mount a magnifying glass of high power — ^the positive eye-piece of a telescope serves the purpose admirably — and adjust it so that the gauze is accurately in focus. To obtain the coincidence of the image of the first gauze with the second, we have now only to move 348 Practical Physics. [Ch. XIIL $ 53. the stand, carrying the second gauze and ms^nifying glass, until the image also comes accurately into focus. The difference of direction of the wires prevents any confusion of the images. A lamp should be put behind one of the gauzes to in« crease the illumination, and care taken that the brightest part of the flame, the object, the centre of the lens, and the screen are in the same straight line. A special case of the foregoing is sometimes used for determining the focal length of a lens. From the formula ^ 4/ we see that if a =0, Le. if the two positions of the lens coincide, then/= >, or one quarter of the distance between 4 the screens. When this b the case the quantity / is at its minimum value ; for solving the equation for / we get The quantity / being the distance between the screens is essentially positive, so that the root with the negative sign gives no applicable result, hence the smallest value ad- missible is /= 2/+ >/4/^ which occurs when a ^o, Le. In this case u^v^ox the image and object are at equal distances from the centre of the lens, and therefore the image is the same size as the object This last property may be used to determine the focal length, by using as object a scale engraved on glass and as screen another such scale ; adjust the lens and receiving scale so that for a par- ticular coloured light the divisions of the image exactly correspond with the divisions of the scale on which it is received. Measure the disUmce of the screens apart, and divide by four, and we get the local length of the lens. Ch. Xin. § 53.] Mirrors and Lenses. 349 A magnifying glass should be used to observe the image, and the observation^ as usual, repeated several times. We know that the focal length of a lens depends on the refractive index of the material of which it is composed, and that this is different for the different rays of the spectrum, so that we should expect to get different values for the focal length by iUuminating the object with differently coloured rays. The methods just described for finding the focal length enable us to do this by placing between the lamp JEuid the object plates of variously coloured glass, red, green, or blue, for example. The position of the receiving screen and consequent value of the focal length win differ in the three cases. Observations with the blue glass will present, perhaps, the greatest difficulty, for most blue glasses let through some red light as well, so that two images are formed a little way apart, one for the blue and the other for the red light If^ then, we are using the wire grating as object, the spaces, when focussed for blue light, will appear blue in the image and the wires red, while if we use the same glass in finding the focal length for the red light, we must focus so that the wires look blue and the spaces red It is quite easy to adapt the method of this section for finding accurately the focal length of the lens, taking into account the thickness, as follows : — Since u and v are measured from the principal points, and the distance between these is very nearly ^^^— /, we have or and 3 so Practical Physics. [Ch. XIIL f s^, whence the expression for the focal length becomes and this reduces to ^ V M 4/* ' we have, therefore, to correct our first approximate valne by subtracting the quantity Experiment, — Determine the focal length of the given lens for red, green, and blue light, and verify your results by the modified method. Enter results thus : — Lens A. /-ass. Red . 70-S 58'8 58-6S Green. • 737 58*4 58-27 Blue . 75*8 S^'i 57'8 54. Measurement of the Fooal Length of a Concave Lens. Method I (requiring a more or less darkened room): — Place in front of the lens a piece of black paper with two narrow slits a, a' cut parallel to each other at a known distance apart, and let light which is quite or nearly parallel fall on the lens (fig. 28). Two bright patches will be formed on a screen at a, a*^ by the light passing through the two sUts, and the rays Fio. ■•• forming them will be m the same directions as if they came Ch. XIII. § 54. J Mirrors and Lenses. 351 from the principal focus f of the lens. If then we measure a of and c x, and if c f =/ we have /+CX aoT from which / can be found. The distance between the centres of the bright patches can be measured with a pair of compasses and a finely divided scale, or by using a scale as the screen on which the light falls. In consequence of the indistinctness of the bright patches, this is only a very rouRh method of determining the focal length. Method 2: — The second method consists in placing in contact with the given concave lens a convex lens sufficiently powerful to make a combination equivalent to a convex lens. Let the focal length (numerical) of the concave lens be / that of the auxiliary convex lens /, and that of the com- bination F. Then I _£_i / / ? The values of > and f can be found by one of the methods described for convex lenses. In selecting a lens with which to form the combina- tion it should be noticed that, if f and/ differ only slightly, say by i centimetre, an error of i millimetre in the deter- mination of each, unless the errors happen to be in the same direction, will make a difference of one- fifth in the result The auxiliary lens should therefore be chosen to make the difference f— / as large as possible— Le. the concave lens should with the convex produce a combination nearly equiva- lent to a lens with parallel foces, so that \ may be very nearly equal to ^ 352 Practiced Physics. [Ch. xill. § 54. For greater accuracy the light used should be allowed to pass through a plate of coloured glass, so as to render it more nearly homogeneous. Experiment — Determine by the two methods the focal length of the given lens. Lens D. Enter results thus : — Method I. — Distance between slits . 2*55 cm. Distances between images Distance from lens to screen . 475 « 3300 , Focal length .... 38-24 „ Method 2. — Focal length of convex lens . Focal length of combination . Focal length required 29*11 cm. 116-14 H 38-85 n P. Focal Lengths. Additional Methods of MeasuremeiLt Other methods for measuring focal lengths depending on various properties of lenses and mirrors have been devised. Thus, consider a source of light on the axis of a con- vex lens, so placed that a real image of the source is formed on the other side of the lens. If the light fall on a plane mirror, it will be reflected back through the lens, and form an image real or virtual, as the case may be. If the mirror be placed so that the image formed by the lens falls on the mirror, the light will be reflected back, and a real image of the source will be formed coincident with the source itselfl In general this will only happen for one position of the mirror ; but suppose the object is at the principal focus of the lens, then the rays from any point on the object form a parallel pencil on fiaUing on the mirror. They will there- fore be reflected as a parallel pencil from the mirror what- ever be the distance between it and the lens, and will again be brought to a focus at the same distance from the lens as the object Thus, if an image be formed at the same distance from the lens as the object, and if this image is not alt^ed by shifting the mirror, keeping its plane normal to the axis of the leny, we know that the object is at the principal focus Ch. Xin, § p.] Mirrors and Lenses. 353 of the lens, and the distance between the object and the lens is the focal length. The image in this case is inverted. Fig. XXX shews Ficxm. the paths of the rays. To perform the experiment, : place a pin in a clip, having ad- justed the lens and mirror so that the axis of the lens is approxi- ; mately normal to . the mirror, and \ move the pin ; about until, look- \ ing at it from some little distance, the image of the pin is seen, as in § 49, to coincide with the pin Then measure the distance pa between the pm and the lens. The convex lens and plane mirror are equivalent to the concave mirror {cf. § 49). A similar method may be used with a concave lens and e mirror to find the focal length of the lens. Light diverging from an object Q (fig. xxxi) is allowed, after refraction through a concave lens, to fall on a concave mirror. It is reflected from this, and converges after re- 354 Practical Physics. [Ch. XIIL f P. flexion towards a point ^, which is the real image in the mirror of ^, the image of Q formed by the lens. But before reaching ^ the light again falls on the lens and is refracted by it to q', at which point a real image of q is formed. If the distances of Q and q' from the lens be observed, and if the focal length of the mirror and the distance between it and the lens be known, then the focal length of the lens can be found. The simplest case is that in which q and q' coincide. When this happens it is clear that the light after reflexion retraces its path ; it falls normally on the mirror. Thus q and g^ coincide at th^ centre of the mirror, and if r be the radius of the mirror and c a = <z, then c^ = r — a, and we have c^ CQ 7' •• /""r-fl cq' and by observing c Q we can find /, the focal length of the concave lens. 55. Focal Lines. When light falls obliquely on a convex lens a refracted pencil does not converge to a point, but to two focal lines in planes at right angles. Let us suppose the lens placed normal to the incident light which is travelling in a horizontal direction, and then turned about a vertical axis till the angle of incidence is 0, and the angle of refrac- tion 0', then the primary focal line is vertical, the secondary is horizontal, and if u be the distance of the source of light from the lens, Z',, v^^ the distances of the focal lines, supposed to be real, and / the focal length of the lens, we have ' j^ , £ ft cos <^' — cos <^ I Vx u (ft— 1) cos' <^ /* J. 4. ^ — Mcos </>^— cos 9 I , v^ u ft— I y' » See Parkinson's 0//ics, p. loi. The signof « has been changed. Ch. Xlil. § 55.] Mirrors and Lenses. 355 . »1 » I • • — v^ u sec' ^ = ~cos' W + Z'a If, then, we determine v^ and z^j, this equation will give us the value of <^ and if the apparatus can be arranged so that ^ can readily be measured, the comparison of the value given by the formula with the result of the measurement enables us to check the formula. To measure ^ the stand carrying the lens should be capable of rotation about a vertical axis, and a horizontal circle attached to it so that its centre is in the axis. A pointer fixed to the moving part of the stand turns over the circle. The reading of the pointer b taken when the lens is placed at right angles to the light, and again when it has been placed in the required position. The difference between the two gives the angle of incidence. To find Vx and v^ it is best to use as object a fine illuminated pin-hole, and to receive the light after traversing the lens on a screen of ground glass with the roughened side to the light. For one position of the screen the vertical lines will appear to be distinctly focussed, while the horizontal are hardly visible. The screen then is in the position of the primary focus, and the distance between it and the Ions is v^* For a second position of the screen the horizontal lines are in focus and the vertical are not seen. This gives the secondary focus, and we can thus find v^. Each observation will require repeating several times, and in no case will the images formed be perfectly clear and well-defined. Avery good result may, however, be obtained by using the homogeneous light of a sodium fiame behind the gauze, and receiving the image upon a second gauze provided with a magnifying lens, as described in § 53. A A a 3S6 Practical Physics. (Ch. XIIL $ 5$. Experiment — Light falls obliquely on a lens ; determine the position of the primary and secondary foci, and hence, find the angle of incidence. Enter results thus : — u « 102 ; Vj - 1 20 ; V, a Zy, Hence cos'^ = -83, * « 24° 39'. Q. Focal Lines formed by a Prism. When light diverging from a point falls on a prism, the emergent light diverges from two focal lines. If the tA^ of the prism be vertical, and if the axis of the incident pencil be at right angles to the edge, the focal lines are horizontal and vertical The position of the horizontal focal hne is independent of the angle of incidence ; that of the vertical focal line changes as the incidence is varied. The vertical line is known as the primary focal line, the hori- zontal line as the secondary. If u be the distance of the object from the prism, which we suppose to be thin, and Vi, v^ the distances of the primary and secondary focal lines, then it is shewn (Parkinson's * Optics,' p. 88) that cos^ 0' cos* ^ cos* cos* ^ * where 0, ^^f are angles of incidence and emergence. If Uie light after passing through the prism fall on a suitable convex lens, real images of the focal lines are fonned by the lens. Thus, in fig. xxxii, o is the source of light ; ^,, ^, the two focal lines formed by the prism a ; Q], q, the real images of these formed by the lens c. These images Qi, Q2 might be received on a screen ; it is better to look at them from behind with an eye-piece— an ordinary watchmaker's glass will do, though a Ramsden's eye- piece with cross-wires set at 45® to the horizon is preferable. If the focal length of the lens c be known, and the dis* tances c q„ c Qj be measured, the values of Vx and v^ can Ch. xiii. § Q.] Mirrors and Lenses. 357 be calculated, and then by measuring the angles of incidence and emergence the formula can be verified. In perfonning the experiment it is best to use for the source o a wire gauze, the wires being set vertically and Fic uiiL horizontally. This is illuminated by a Bunsen burner with a sodium flame. In the position of the primary focal line distinct images of the vertical wires will be formed ; in the position of the secondary line the horizontal wires will be seen clearly. If the position ol the prism be that of minimum deviation, so that ^ = if*, then we shall have »i = jij = ». Thus f I and ^;, and therefore Q, and Q„ coincide, and if the eye-piece be focussed on the image both vertical and hori- zontal wires will be seen. If now the angle of incidence be changed, the vertical wires will become indistinct, while the others remain clear, shewing that the position of the secondary focus is independent of the angle oi incidence. On drawing the eye-piece back or pushing it forwards, as the case may be, a badly defined image of both sets of wires, corresponding to the position of the circle of least con- fusion, comes into view, while on moving the eye-piece still further in the same direction the horizontal wires disappear. 358 Practical Physics. [Ch. XIIL § Q. but the vertical wires are seen sharply defined as a set of vertical bars against a uniform field. Experiment— Sh^^ that the primary and secondary focal lines formed by a prism coincide when the deviation of the prism is a mininum, and measure the distance between their images formed by a convex lens when the prism is turned lo* from this position. On the Measurement of Magnifying Powers of Optical Instruments, The magnifying power of any optical instrument is the ratio of the angle subtended at the eye by the image as seen in the instrument to the angle subtended at the eye by the object when seen directly. If the object to be seen is at a short distance from the eye, and the distance can be altered, the eye must always be placed so that the object is at the dis- tance of most distinct vision (on the average, 25 cm.) ; and any optical instrument is focussed so that the image seen is at the distance of most distinct vision. Thus the magnifying power of a lens or microscope is the ratio of the angle sub- tended at the eye by the image in the instrument to the angle subtended at the eye by the object when placed at the distance of most distinct vision. Telescopes are, however, generally used to observe objects so distant that any alteration which can be made in the distance by moving the eye is very small compared with the whole distance, and hence for a telescope the magnifying power is the ratio of the angle subtended by the image in die telescope to the angle subtended by the object Then again this image is at the distance of distinct vision for the eye, but the focal length of the eye-piece is generally so short that the angle subtended by the image at Uieeye is practically the same as if the eye-piece were focussed so that the image was at an infinite distance. Thus suppose the small image / q (fig. 29X formed by the objea-glass a, is in such a position with reference to the Ch. XIII. § 56.J Mirrors and Lenses. 359 eye-piece b that the image of it p'q' formed by the eye- piece is at the same distance as the object p q. Since the object is very distant the angle subtended by it at the centre a of the object-glass, which is equal to the angle/ a f, is practically the same as that subtended by it at the eye, and the angle subtended by the image at the eye is practically the same as the angle /^^. These angles being very small, they will be proportional to their tangents, and the magnifying power wiU be equal to either (i) the ratio of the focal length of the object-glass Fig. 99. 0?' to the focal length of the eye-piece ; or (2) the ratio of the absolute magnitude (diameter) of the image p' q' to that of the object pq when the telescope is so focussed that these two are at the same distance from the eye. On this second definition of the magnifying power depends the first method, described below, of finding the magnifying power of a telescope. 56. Heasuiement of the Magnifying Power of a Telescope. — First Method. Place the telescope at some considerable distance from a large scale, or some other well-defined object divided into a series of equal parts— the slates on a distant roof, for example. Then adjust the eye-piece so that the image 36o Practical Physics. [Ch. xiil § 56. seen in the telescope coincides in position ^ith the scale itsel£ In doing this, remember that when the telescope is naturally focussed the image is about ten inches ofT ; and as the eye-piece is pulled further out, the image recedes until the small image formed by the object-glass is in the principal focus of the eye-glass, when the image seen is at infinity. The required position lies between these two limits, and is attained when the image seen through the telescope with the one eye is quite distinct, while at the same time the scale, as seen directly, is distinctly seen by the other eye looking along the side of the telescope ; Fio. 90. and, moreover, the two do not appear to separate as the eyes are moved from side to side. Then the appearance to the two eyes IS as sketched in fig. 30, where the magni- fying power is about 8. The number of divisions of the scale, as seen directly, covered by one of the divisions of the image of the scale can be read ofi^ and this gives evidently the ratio tof the tangents of the two angles, / ^ ^, p a Q, and hence the magnifying power of the telescope. If the scale used be in the laboratory, so that its distance from the telescope can be measured, the experiment should be made at different distances. Instead of reading the number of divisions of the scale occupied by one division of the image, it is best to count those occupied by some six or eight divisions of the image and divide one number by the other. Experiment. — Determine, at two different distances, the magnifying power of the given telescope. Enter resulu thus : — Telescope No. 3. Distance between scale and telescope 1000 COL Ch. XIIL § 56.] Mirrors and Lenses. 361 Lower edge of image of division 76 is at o on scale. Lower edge of image of division 69 is at 99 on scale. Magnifying power - 99^9. =14-14 76 — 69 Distance ■■ 500 cm. Lower edge of image of division 72 is at 95. Lower edge of image of division 78 is at 3. Magnifying power = p"^ "I5*3 78-72 57. Heasnremeiit of the Magnifying Power of a Telescope.— Second Method. The magnifying power of a telescope for an infinitely distant object may be taken as the ratio of the focal length of the object-^lass to that of the eye-piece, and may be found by the following method : — Focus the telescope for parallel rays as follows : — (i) Focus the eye-lens by sliding in the socket until the cross-wires are seen distinctly. (2) Direct the telescope to the most distant object visible from an open window— a vane is generally a con- venient object — and move the eye-piece and cross-wires together as one piece (there is generally a screw for doing this, but sometimes it has to be done by pulling out the tube by hand) until the distant object is clearly seen as well as the cross-wires, and so that there is no parallax, i.e. so that on moving the eye across the aperture of the eye-piece the cross-wires and image do not move relatively to each other. This will be the case when the image of the distant object formed by the object-glass is in the plane of the cross-wires. The telescope is then said to be focussed for infinity or for parallel rays. Next, screw off the cover of the eye-piece — without altering the focus — and screw out the object-glass and substitute for it an oblong-shaped diaphragm, the length of which must be accurately measured : let it equal i- The 362 Practical Physics. [Ch, XIIL § S7- measurement can be easily effected by means of a pair of dividers and a fine scale. The distance of this from the optical centre of the eye- piece is F+yj F and / being the foc^ lengths of the object- glass and eye-piece respectively. An image of this oblong aperture will be formed by the eye-piece at a distance v on the other side of its optical centre, where I I I /"^fTJ"^^" Now measure the length of this image by bringing up to it a micrometer scale engraved on glass, such as is made for use in a microscope, graduated to tenths of a milli- metre and having a lens mounted in front of it to &cilitate the reading. Place the micrometer in a clip, and adjust the height and distance until the scale and the image of the aperture are both distinctly seen on looking through the lens attached to the micrometer. In this way the length of the image of the diaphmgm can be determined in terms of millimetres and tenths. Let this be /. Of course any other convenient form of micrometer may be used for this purpose. Fig. 3z. f l Then the magnifying power 2 = 7 For if ll' (fig. 31) be the diaphragm aperture, so that L l' = L, M the micrometer, and //' the image formed, c the optical centre of the eye-piece, tlien l_ll'_cl f 4" / / //' 0/ V But Ch^ XIII. § 57.] Mirrors and Lenses. 363 In measuring the length of the image by the micro- meter scale, the aperture should not be too brightly illu- minated, or the image may be blurred and mdistinct The telescope should on this account be pomted at a sheet of grey filter-paper or other slightly illuminated uniform sur- face, giving just light enough for reading the micrometer scale. Experiment, — Determine the magnifying power of the given telescope. Enter results thus:— Telescope No. 2. Length of aperture .... 218 cm. Length of image . . . . -16 cm. Magnifying power .... 13-6 58r Meafuremant of the Magnifying Power of a Lens or of a MicroBoope. A lens or microscope is used for the purpose of viewing objects whose distance from the eye is adjustable, and in such cases the magnifying power is taken to be the ratio of the angle subtended at the eye by the image as seen in the mstrument to the angle subtended at the eye by the object when placed at the distance of most distmct vision (generally 25 cm.). The instrmnent is supposed to be focussed so that the image appears to be at the distance of most distinct vision. The method described for a telescope in § 56 is applicable, with slight alteration, to the case of a lens or microscope. The instrument is focussed on a finely divided scale ; one eye looks at the magnified image while the other looks at another scale placed so as to be 25 cm. away from the eye, and to appear to coincide in position with the image of the first scale viewed through the instrument Suppose the two scales are similarly graduated, and that x divisions of the magnified scale cover x divisions of the scale seen directly, then the magnifying power is x/x If the two scales be not 3^4 Practical Physics. [Ch. XIII. % 58. similarly divided — and it is often more convenient that they should not be so — a little consideration will shew how the calculation is to be made. Thus, if the magnified scale be divided into m^ of an inch, and the unmagnified one into rfi^^ and if x divisions of the magnified scale cover x un- magnified divisions, then the magnified image of a length of x\m inches covers an unmagnified length of x//r inches, and the magnifying power is therefore tnyijnx. The following modification of the method gives tbe two images superposed when only one eye is used : — Mount a camera-lucida prism so that its edge passes over the centre of the eye-lens of the microscope. Then half the pupil of the eye is illuminated by light coming through the micro- scope, and the other half by light reflected at right angles by the prism. If a scale be placed 25 cm. away from the prism, its image seen in the camera-lucida may be made to coin- cide in position with the image of the scale seen by the other half of the pupil through the microscope. To make this experiment successful, attention must be paid to the illumination of the two scales. It must be re- membered that magnifying the scale by the microscope reduces proportionately the brightness of the image. Thus the magnified scale should be as brightly illuminated as possible, and the reflected scale should be only feebly illumi- nated. It should also have a black screen behind it, to cut off the light from any bright object in the background. A piece of plane unsilvered glass set at 45% or a mirror with a small piece of the silvering removed, may be used in- stead of the camera-lucida prism. The magnifying power of a thin lens may be calculated approximately from its focal length. The eye being placed close to the lens, we may take angles subtended at the centre of the lens to be equal to angles subtended at the eye. Now a small object of length / placed at a distance of 25 cm subtends an angle whose measure may be taken to be 7/25. UTien the lens is interposed the linage is to be at a distance Ch. XIIL § 58.1 Mirrors and Lenses. 3^5 of 25 cm., and the distance between the object and eye must be altered ; the object will therefore be at a distance u where u 25""/' The angle subtended by the image is similarly measured by its length divided by 25, and this is equal to //«, or (7^^) Thus the magnifying power is I 25 or y+i. / A microscope with a micrometer scale in the eye-piece is sometimes used to measure small distances. We may there- fore be required to determine what actual length corresponds, when magnified, to one of the divisions of the micrometer scale in the eye-piece. For this pmpose place below the object-glass a scale divided, say, to tenths of a millimetre, and note the number of divisions of the eye-piece scale which are covered by one division of the object scale seen through the micro- scope ; let it be a. Then each division of the eye-piece scale corresponds clearly to i/a of one-tenth of one milli- metre, and an object seen through the microscope which appears to cover b of these eye-piece divisions is in length equal to b\a of one-tenth of a millimetre. If we happen to know the value of the divisions of the eye-piece scale we can get from this the magnifying power of the object-glass itself, in the case in which the microscope b fitted with a Ramsden's or positive eye-piece, and thence, 366 Practical Physics. [Ch. XIIL § 5! on determining the magnifying power of the eye-piece, find that of the whole microscope. For if f«i be the magnifying power of the object-glass, tn^ that of the eye-piece, then that of the whole microscope is ^i x m^. Thus, if the eye-piece scale is itself divided to tenths of millimetres, since one-tenth of a millimetre of the object scale appears to cover a tenths of a millimetre of the eye- piece scale, the magnifying power of the object-glass is a. If, on the other hand, the microscope is fitted with a Huyghens or negative eye-piece, then the eye-piece scale is viewed through only the second or eye lens of the eye piece, while the image of the object scale, which appears to coincide with it, is that formed by reflection at the object-glass and the first or field lens of the eye-piece ; the magnifying power determined as above is that of the com- bination of object-glass and field lens. To determine the magnifying power for the whole microscope, in this case we must find that of the eye-lens and multiply the two together. It should be noticed that the magnifying power of a microscope depends on the relative position of the object- glass and eye-piece. Accordingly, it the value of the mag- nifying power is to be used in subsequent experiments, the focussing of the object viewed must be accomplished by moving the whole instrument. Experiment,— Ti^X.txTCiWi^ by both methods the magnifying power of the given microscope. Enter the results thus: — First method. — Scale viewed through microscope graduated to half-millimetres. Scale viewed directly graduated to milli- metres. Three divisions of scale seen through microscope cover 129 of scale seen directly. Magnifying power - ^2 « 86. Ch. XIII. § 58.) Mirrors and Lenses. ^ 367 Second method. — One division of eye-piece scale » *5 mm. Three divisions of scale viewed cover 14*57 divisions of eve- piece scale. Magnifying power of eye-piece 18. /. Magnifying power of microscope - — ^, - x 18 « ft7'4. 59. The Testing of Plane Surfaces. The planeness of a reflecting surface can be tested more accurately by optical means than in any other way. The method depends on the fact that a pencil of parallel rays remains parallel after reflexion at a plane surface. To make use of this, a telescope is focussed on a very distant object — so distant that the rays coming from it may be regarded as parallel. The surface to be tested is then placed so that some of the parallel rays from the distant object fall on it and are reflected, and the telescope is turned to receive the reflected rays — to view, that is, the reflected image. If the surface be plane, the reflected rays will be parallel and the image will be as far away as the object When viewed through the telescope, then, it will be seen quite sharp and distinct. If, on the other hand, the surface be not plane, the rays which enter the object- glass will not be parallel, and the image seen in the tele- scope will be blurred and indistinct We can thus easily test the planeness of a surface. If the surface is found to be defective, the defect may arise in two ways : — {a) From the surface being part of a regular reflecting surface — a sphere or paraboloid, for example — and not plane. In this case a distinct image of the distant object is formed by reflexion at the surface ; but, the surface not being plane, the pencils forming the image will not be pa- rallel, and therefore, in order to see it, we must alter the 368 Practical Physics, [Ch, XIIL ; 59* focussing of the telescope. We shall shew shortly how, by measuring the alteration in th^ position of the eye-piece of the telescope, we can calculate the radius of curvature of the surface. (b) In consequence of the general irregularity of the surface. In this case we cannot find a position of the eye- piece, for whicli we get a distinct image formed — the best image we can get will be ill-defined and blurred. We may sometimes obtain a definite image by using only a small part of the reflecting surface, covering up the rest 111 is may happen to give regular reflexion, and so form a good image. To test roughly the planeness of a surface or to measure its curvature, if the latter be considerable, an ordinary ob- ser\'ing telescope may be used. Focus it through the open window on some distant, well-defined object A vane, if one be visible, will be found convenient Place the surface to reflect some of the raj-s from the distant object at an angle of incidence of about 45^, and turn the telescope to view the reflected image. If the image is in focus, the surface is plane. If by altering the focus we can again get a well-defined image, the surface reflects regularly, and is a sphere or something not differing much from a sphere ; if the image can never be made distinct and clear, the surface is irregular. I^t us suppose we find that by a slight alteration in the focus we can get a good image, we shall shew how to measure the radius of curvature of the surface. To do this accurately, we require a rather large telescope with *an object-glass of considerable focal length, say about i metre. It will be better, also, to have a collimator. This con- sists of a tube with a narrow slit at one end of it and a convex lens at the other, the focal length of the lens being the length of the tube ; the slit is accordingly in the prmd- pal focus of the lens, and rays of light coming fi*om it are rendered parallel by refraction at the lens. Sometimes a tube carrying the slit slides in one carrying the lens, so that the distance between the two can be adjusted. Cif. XIII. 5 59.] Mirrors and Lenses. 369 We shall suppose further that there is a distinct mark on the telescope tube and another on the sliding tube to which the eye-piece is attached. We shall require to mea- sure the distance between these marks ; the line joining them should be parallel to the axis of the telescope. The telescope should also be furnished with cross-wires. Focus the eye-piece on the cross- wires. Turn the tele- scope to the distant object and adjust the focussing screw, thus moving both eye-piece and cross- wires relatively to the object-glass, until the object is seen distinctly and without any parallax relatively to the cross-wires. To determine when this is the case move the eye about in front of the eye-piece and note that there is no relative displacement of the image and the cross-wires. Measure with a millimetre scale, or otherwise, the dis- tance a, say, between the two marks on the telescope tubes. Repeat the observation four or five times. Take the mean of the distances observed and set the instrument so that the distance between the marks is this mean. - Now point the telescope to the collimator, place a lamp behind the slit of the latter, and adjust the distance between the slit and the lens until the slit appears to be properly focussed when viewed through the telescope. When this is the case the rays issuing from the collimator lens are accurately paralld. Place the reflecting surface to reflect at an angle of in* ddence of about 45^ the light from the collimator, and turn the telescope to view it When the reflecting material is transparent and has a second stuface nearly parallel to the first, the light reflected from it will form an image which will be visible and may cause inconvenience ; if this be so, cover the second surface with a piece of wet coloured blotting-paper We require to know the angle of incidence. To find this accurately it would be necessary to use for the collimator the collimator of a spectrometer and to mount the surface BB 370 Practical Physics. [Ch. XIIL § s> on the table of the spectrometer. The angle then could be found as described in § 62. For most purposes, howeva; the angle of incidence can be found by some simpler means^ e.g. by setting the telescope and collimator so that their axes are at right angles, determining when this is the case by eye or with the help of a square, and then placing the sur&ce so as to bring the reflected image of the slit into the field of view ; the angle required will then not differ much from 45^ Let us call it <^ The image seen will not be in focus, but it can be rendered distinct by altering the position of the eye-piece of the telescope. Let this be done four or fi?c times, and measure each time the distance between the two marks on the telescope tubes ; let the mean value be b. Observe also the distance c between the object-glass and the reflecting surface, thb distance being measured parallel to the axis of the telescope. Let f be the focal length of the object-glass, ^ the angle of incidence, then r the radius of curvature of the reflecting face is, if that face be convex, given by the formula {b^a) cos ff> For let A B (fig. 32) be a ray incident obliquely at b at an angle ^ a' b' an adjacent parallel ray ; after reflection they will diverge from a point Q behind the surface, and falling on the ob- ject-glass c be brought to a focus at ^, there forming a real image of the distant object, which is viewed b^ the eye-piece d. Let Q ^^^ I ^\.^>>A ^ ^^ ^^ principal fo- cus of the object-glass. Then when the distant object was viewed direcdy, the image fomred by the object-glass was at f, and if d' be the posi- Fig. 38. Ch. XIII. § 59.] Mirrors and Lenses. 371 tion of the eye-piece adjusted to view it, we have d'f = d^, and hence f ^ = d d', but d d' is the distance the eye-piece has been moved ; hence we have rq^h—a^ and cr = f; Also c B = r, and since q is the primary focaMine * of a pencil of parallel rays incident at an angle <^ B Q = ^ R cos ^ ; •*. CQ = ^+iRCOS^ But CQ Qq F and J+Jrcos^ f t-^-h—a *^^a)cos^ In the case of a concave surface of sufficiently large radius it will be found that d is less than a ; the eye-piece will re- quire pushing in instead of pulling out ; and the radius of curvature is given by the formula (fl — ^)cos<^ We have supposed hitherto that the slit is at right angles to the plane of reflexion, and the primary focus, therefore, the one observed. If the slit be in the plane of reflexion > See Parkinson's Opius (edit 1870), p. 6a BB2 372 Practical Physics. [Cii. XIIL $ 59L the image seen will be formed at the secondary focal line, and the formula will be (a— ^)sec^ * a, 3, c^ &c, having the same meaning as before. Again let us suppose that the plate of material examined has two faces, each of which has been foimd to be plane. We can use the method to determine if they are pualld, and if not to find the angle between them. For make the adjustments as before, removing, however, the wet blotting paper from the back face. If the two faces be strictly parallel only one image of the slit will be seen, for the rays firom the front and back surfaces will be parallel after reflexion. If the faces be not parallel, two images of the slit will be seen. Let us suppose that the angular distance between the two images can be measured either by the circle reading of the spectrometer, if the spectrometer telescope is being used, or by the aid of a micrometer eye-piece if that be more convenient ; let this angular distance be d ; then the Fio. 33. angle between the faces is given bjr the equation . __ D cos aficos^'* where ^ is the angle of refractios corresponding to an angle of in- cidence ^ and \jl the refractive index of the material ; d and / are supposed so small that we may neglect their squares. For (fig. 33) let A B c,A DE be the two 6ces of the prism, pbq, PBDCQ'tbe paths of two rays ; Jet q b, 0' c meet in o» then q o q' ^ d, B A D«> il Ch. XITl. S 5^j Mirrors and Lenses, 373 Hence d = qoq' = OB a— oca = \ IT — ^— OCA, /. OCA = ^ir — ^ — D. Again DCA = E DC — t'^ ADB — I = D EC — 2 f s=^ir— ^'— 2l' Also since d c and c q' are the directions of the same ray inside and outside respectively, cos OCA = ft cos DC a; /. sin(^ + d) = /I sin(^' + 2 /) ; •*• sin ^+Dcos^ss /i(sin^' + a/cos^'), neglecting d* and P. But sin ^ = /i sin if/ ; .__ DCOS ^ 2/i cos ^^ Again, it may happen that one or both faces of the piece of glass are curved ; it will then act as a lens, and the following method will give its focal length. The method may be advantageously used for finding the focal length of any long-focussed lens. Direct the telescope to view the collimator slit, and focus it; interpose the lens in front of the object-glass. The focus of the telescope will require altering to bring the slit distinctly into view again. Let us suppose that it requires to be pushed in a distance X. Let c be the distance between the lens and the object- glass of the telescope, then the parallel rays from the colli- mator would be brought to a focus at a distance/ behind the lens, Le. at a distance /— r behind the object-glass ; they (all, however, on the object-glass, and are brought by it to a focus at a point distant f— :c from the glass. • • 374 Practical Physics. [Ch. XIII. S 59. and firom this we find If the lens be concave, the eye-piece of the telescope will require pulling out a distance x suppose ; and in this case the rays falling on the object-glass wUl be diverging from a point at a distance /-fr in front of it, and will conveige to a point at a distance F-f x behind it .•/ ^ . We infer, then, that if the eye-piece requires pushing in the lens is convex, and if it requires pulling out it is concave. Moreover, we note that all the above formulae both for reflexion and refraction are simplified if F = r; that is to say, if the distance between the object-glass and the reflect- ing surface or lens, as the case may be, is equal to the focal length of the object-glass. If this adjustment be made, and if j; be the displace- ment of the eye-piece in either case, we have for the radius of curvature of the surface R = 2F* X cos <^ and for the focal length Experiments, (i) Measure the curvature of the faces of the given piece of glass. (2) If both faces arc jslane, measure the angle between them. (3) If either face is curved, measure the local length ot U)e lens formed by the glass. Ch. XIIL § 59.1 Mirrors and Lenses. Enter results thus : — (1) Scale used divided to fiftieths of an inch. Angle of incidence 45**. First face, concave. . 175 177 17-5 375 Values of a Values of b Value of a -^ Values of c 3.9 3.9 3.8 12*9 Focal length of object-glass Value of R (2) 4, -45^ /i -1*496 <^'- 28^12' i - 1'32'' F -54 cm. n - 2-35 „ / = i^-97„ (3) 17-65 17-6 cm. Mean 17*59 » 3-8 3-8 .. Mean 3-84 ^ • 1375 „ 13*2 13-0 Mean 13*03 • 54*3 . 2487 CHAPTER XIV. SPECTRA, REFRACTIVE INDICES, AND WAVE-LENGTHS. A BEAM of light generally consists of a combination of differently-coloured sets of rays ; the result of the decom- position of a compoimd beam into its constituents is called a spectrum. If the beam be derived from an illuminated aperture, and the spectrum consist of a series of distinct images of the aperture, one for each constituent set of rays of the compound light, the spectrum is said to be pure. A spectroscope is generally employed to obtain a pure fpectnim. The following method of projecting a pure 376 Practical Physics, [Ch. xrv. § 59. spectrum upon a screen by means of a slit, lens, and prism, illustrates the optical principles involved The apparatus is arranged in the following manner. The lamp is placed at l, fig. 34, with its flame edgewise to the slit ; then the slit s and the lens m are so adjusted as to give a distinct image of the slit at s' on the screen a b; the length of the slit should be set vertical. The prism p q r is then placed with its edge vertical to receive the rays after passing through the lens. All the rays from the lens should Fig. 34. .--'L fall on the front face of the prism, which should be as near to the lens as is consistent with this condition. The rays will be refracted by the prism, and will form a spectrum a' b' at about the same distance from the prism as the direct image s'. Move the screen to receive this spectrum, keeping it at the same distance from the prism as before, and torn the prism about until the spectrum formed is as near as possible to the position of s', the original image of the slit ; that is, until the deviation is a minimum. The spectrum thus formed is a pure one, since it contains an ima^gt; CiL XIV. 1 60.] Spectra^ Refractive Indices, &c, 17 j of the slit for every different kind of light contained in the incident beam. 60. The Spectroscope* Mapping a Spectrum, We shall suppose the spectroscope has more than one prism. Turn the telescope to view some distant object through an open window, and focus it In doing this adjust first the eye-piece until the cross-wires are seen distinctly, then ^move the eye-piece and cross-wires by means of the screw until the distant object is clear. The instrument should be focussed so that on moving the eye about in front of the eye-lens no displacement of the image relatively to the cross-wires can be seea Remove the prisms, and if possible turn the telescope to look directly into the collimator. Illuminate the slit and focus the collimator until the slit is seen distinctly. Replace one prism and turn the telescope so as to receive the refracted beam. ' Turn the prism round an axis parallel to its edge until the deviation of some fixed line is a mini- mum (see § 62, p. 391). For this adjustment we can use a Bunsen burner with a sodium flame. If the prism have levelling screws, adjust these until the prism is level. To test when this is the case fix a hair across the slit, adjusting it so that when viewed directly it may coincide with the horizontal cross-wire of the eye-piece. The hair will be seen in the refracted image cutting the spectrum horizontally. Adjust the levelling screws of the prism until this line of section coincides with the cross- wire. In some instruments the prisms have no adjusting screws, but their bases are ground by the maker so as to be at right angles to the edge. Having placed the first prism in position, secure it there 378 Practical Physics. [Ch. xn\ § 6a with a clamp, and proceed to adjust the second and other prisms in the same way. The table of the spectroscope is graduated into d^rees and minutes, or in some instruments there is a third tube carrying at one end a scale and at the other a lens whose focal length is the length of the tube. The scale is illu- minated from behind by a lamp and is placed so that the rays which issue from the lens fall on the face of the prism nearest the observing telescope, and being there reflected form an image of the scale in the focus of the telescope. Bring the vertical cross-wire, using the clamp and tan- gent-screw, over the image of the slit illuminated by the yellow sodium flame and read the scale and vernier, or note the reading of the reflected ^:ale with which it coincides. Replace the sodium flame by some other source of light the spectrum of which is a line or series of lines, as, for example, a flame coloured by a salt of strontium, lithium, or barium, and take in each case the readings of the reflected scale or of the vernier when the cross-wire coincides with the bright lines. Now the wave-lengths of these lines are known; we can therefore lay^down on a piece of logarithm paper a series of points, the ordinates of which shall represent wave-lengths, while the abscissae represent the graduations of the circle on scale. If we make a sufficient number of observations, say from ten to fifteen, we can draw a curve through them, and by the aid of this curve can determine the wave-length of any unknown line ; for we have merely to observe the reading of the circle or scale when the cross- wire is over this line and draw the ordinate of the curve corresponding to the reading observed This ordinate gives the wave-length required.^ In using the diagram or * map ' at any future time we must adjust the scale or circle so that its zero occupies the same position with reference to the spectrum. This can be done by arranging that some well-known line — e.g. D — should » See Glazebrook, Physical Optics^ p. 113. Ch. XIV. 1 6a] Spectra, Refractive Indices, &c. 379 always coincide with the same scale division or circle read- ing. The accuracy of readjustment of the spectroscope should also be tested by comparing the reading of some other well-known line with its original reading. Instead of using the light from a Bunsen burner with metallic salts in the flame, we may employ the electric spark from an induction coil either in a vacuum tube or between metallic points in air. If the vacuum tube be used, two thin wires from the secondary of the coil are connected to the poles of the tube — pieces of platinum wire sealed into the glass. The primary wire of the coil is connected with a battery of two or three storage cells, and on making contact with the commutator the spark passes through the tube. This is placed with its narrow portion close up to and parallel to the slit, and the spectroscope observations made as before. If the spark t>e taken between two metallic poles in air, the two poles placed in the spark-holder are connected with the second- ary and placed at a distance of two or three millimetres apart, and the spark passed between them. ' The spark-holder is placed in front of the sUt, and either the spark is viewed directly or a real image of it is formed on the slit by means of a convex lens of short focus. With this arrangement, in addition to the spectrum of the metal formed by the light from the glowing particles of metal, which are carried across between the poles by the spark, we get the spectrum of the air which is rendered in- candescent by the passage of the spark. The lines will probably be all somewhat faint, owing to the small quantity of electricity which passes at each discharge. To remedy this, connect the poles of the secondary coil with the outside and inside coatings of a Leyden jar, as is shewn in fig. 35. Some of the electricity of the secondary coil is used to charge the jar; the difference of potential between the metallic poles rises less rapidly, so that dis- charges take place less frequently than without the jar; but when the spark does pass, the whole charge of the jar 3lto Practiad Physics. [Ch. xiv. ; eo. passes with it, and it is «)asequently much more brillianL Even with the jar, the sparks pass so rapidly that the im* pression on the eye is continuous.' Tn experiments in which the electric spark is ased, it is well to connect the spectroscope to earth T>y means of a wire from it to the nearest gas-pipe ; this helps to prevent shocks being received by the observer. Sometimes after the spark has been passing for some time it suddenly stops. This is often due to the hammer of the induction coil sticking, and a jerk is sufficient to start it again ; or in other cases it is well to turn the commutator of the coil and allow the spark to pass in the other direction. It may of course happen that the screws regulating the hammer of the coil require adjustment Experiments. Drawa curve of wave-lengthsfor the givenspectroscope, deter- mining the position of ten to fifteen points on it, and by means of it calculate the wave-length of the principal lines of the spec- trum of the given metal. Map the spectnim of the spark passing through the given ' The Intensity of the spark may oftcD be nlficiently incTOsed without the nie of the jar by Iwving ■ lecond imali hiixk ia the ' ciicnit b«*ween A and c aaoM which a spaik («ite*. Ch. XIV. § 6al Spectra^ Refractive Indices^ &c. 381 Comparison of Spectra, Many spectroscopes are arranged so as to allow the spectra of two distinct sources of light to be examined simultaneously. To effect this a rectangular prism abc (fig. 36 [ij) Fig. 36. is placed behind <l the slit of the col- limator in such a way as to cover one half, suppose the lower, of the slit Light coming from one side falls normally on the face BC of this prism, and is totally reflected at the face ab emerging normally from the face ca ; it then passes through the slit LM and falls on the object glass of the collimator. In some cases a prism of 60^ b used (fig. 36 [2]). The second source of light is placed directly behind the slit and is viewed over the top of the prism. One half of the field then, the upper, in the telescope is occupied by the spectrum of the light reflected by the prism, while the other is filled by that of the direct light We may use this apparatus to compare the spectra of two bodies. Suppose we have to determine if a given substance con- tain strontium. Take two Bunsen burners and place in one a portion of the given substance on a piece of thin platinum foil, while some strontium chloride moistened with hydrochloric add IS placed in the other on a similar piece of foil The two spectra are brought into the field. If the strontium lines appear continuous through both spectra, it is clear that the fijrst spectrum is at least in part that of strontium. 382 Practical Physics. [Ch. XIV. § 6a As we have seen already, if we pass a spark in air between metallic poles we get the air lines as well as those due to the metal We may use this comparison method to distinguish between the air lines and those of the metaL For let one set of poles be made of the metal in question, and take for the other set some metal with a simple known spectrum, platinum for example. Arrange the apparatus as described to observe the two spectra. The lines common to both are either air lines or are due to some common im- purity of the two metals ; the other lines in the one spectrum are diose of platinum, in the second they arise from the metal in question. After practice it is quite easy to recognise the distinctive lines of many substances without actual comparison of their spectra with that of a standard Experinunt, — Compare the spectra of the sparks passing between platinum poles and poles of the given metal. Note the wave-lengths of the principal lines in the spark spectrum of the given metal. On Refractive Indices, If a ray of homogeneous light fall on a refracting medium at an angle of incidence ^, the angle of refraction being 4>'f then the ratio sin ^/sin ^' is constant for all values of ^ and is the refractive index for light of the given refrangibility going from the first to the second medium. Let us suppose the first medium is air, then it is not difficult to determine by optical experiments the value of the angle ^ but <^' cannot be determined with any real approach to accuracy. The determination of /w, the refrac- tive index, is therefore generally effected by indirect means. We proceed to describe some of these.' * For proofs of the optical formulae which occur in the succeeding sections, we may refer the reader to Glazcbrook's Physical Optkh chaps, iv. and viii. Ch. XIV. § 61.] Spectra^ Refractive Indices^ &c, 383 6x. Heasorement of the Index of Eefraction of a Plate by means of a Microscope. Let p (fig. 37) be a point in a medium of refractive index /«, and let a small pencil of rays diverging from this point fall directly on the plane-bounding surface of the medium and emerge into air. Let A be the point at which the axis of the pencil emerges, and q a point on pa, such that ap = /aaq3 then the emergent pencil will appear to diverge from q, and if we can measure the distances ap and aq we can find fu To do this, suppose we have a portion of a transparent medium in the form of a plate, and a micro- scope, the sliding tube of which is fitted with a scale and vernier or at least a pointer, so that any alteration in the position of the object-glass when the microscope b adjusted to view objects at different distances may be measured. Place under the object-glass a polished disc of metal with a fine cross engraved on it, and bringing the cross into the centre of the fields focus the microscope to view and read the scale. Repeat the observation several times, taking the mean. Now bring between the metal plate and the object-glass the transparent plate, which, of course, must not be of more than a certain thickness. One surface of the plate is in contact with the scratch on the metal, which thus corresponds to the point p ; the emergent rays therefore diverge from the point Q, and in order that the scratch may be seen distinctly through the plate, the microscope will require to be raised until its object-glass is the same distance from q as it was originally from p. Hence, if we again focus the micro- scope to see the cross, this time through the plate, and read the scale, the difference between the two readings Fig. 37. \ A/ I / li 1 l\ d \\ jl 1 • 1 1 1 1 f 1 1 384 Practical Physics. [Ch. XIV. § 6i. will give U8 the distance p q. Let us call this distance a^ and let / be the thickness of the plate, which we can measure by some of the ordinary measuring apparatus, or, if more convenient, by screwing the microscope out until a mark, made for the purpose, on the upper surface of the plate comes into focus, and reading the scale on the tube. We thus can find pa = /, pq = « But we have AP = /aaq = /*(ap— pq); /. /==,!(/-«). and fi= - — • /—a A modification of this method is occasionally useful for finding the index of refraction of a liquid. Though if the depth of the liquid can be measured it is best to treat it as a plate. Suppose the liquid to be contained in a vessel with a fine mark on the bottom. Focus on the mark through the liquid, and then on a grain of lycopodium dust floating on the surface. If the depth be i/i, the difference between the readings gives us dx l/M. ; let us call this difference a» Then /* Now add some more liquid until the depth is i/i+^s- Focus on the mark again, and then a second time on the floating lycopodium which has risen with the surface ; let the difference between these two be d ; then But the difference between the second and fourth reading, that b to say, of the two readings for the lycopodium grains is clearly the depth of liquid added, so that from these two readings d^ is obtained, and we have fA. o — a Cu. XIV. ( 61.J Sfeara, K,/maivc India,, &-c. jgj £"^/*«M/.— Detennine the index of refraction fil of thp Enter results thus:— Index of refraction of water. <0 fI2 4-54 456 '■333 ■95 4-41 4-65 '■344 ■68 4-07 4-56 ■"345 •38 376 453 '■340 ■43 di6 Mean value of J 4iW 1-319 ■ 1-336 6a. The 8peotromet«r. This instrument (fig. 38) consUts of a graduated cirele generallyfixedmahorizontalposiUoa Acollimatorisrigidlv attached to the circle. The axis of the colUmator is in a plane parallel to that of the circle, and is directed to a point vertically above the centre of the circle. A movable arm, fitted with 3 clamp and tangent screw, carries an astrono- mical telescope which is generally provided with a Ramsden's eye-piece and cross-wires. The position of the telescope with reference to the circle is read by means of a vernier.' Above the centre of the circle there is a horizontal table, which is generally capable of rotation about the vertical axis of the ' See FrDntitpKce,figs.5 and 6, 386 Practical Physics. {Ch. XIV. § 62. circle ; and this table has a vernier attached to it, so that its position can be determined. The whole instrument rests on levelling screws, and the telescope and collimator are held in their positions by movable screws, so that their axes can be adjusted till they are parallel to the circle. On the Adjustment of a Spectrometer, The line of collimation, or axes of the telescope and collimator should lie in one plane, and be always perpen- dicular to the axis about which the telescope rotates. To secure absolute accuracy in this is a complicated problem. In practice it is usually sufficient to assume that the axes of the telescope and collimator are parallel to the cylindrical tubes which carry the lenses. Level the table of the instrument by means of a spirit-level and the levelling screws, afterwards level separately the telescope and colli- mator by means of a level and the set screws attached to each. The axis of each is now parallel to the plane of the circle. See that the clamp and tangent screw work properly, and that the instrument is so placed that the vernier can be read in all positions in which it is likely to be required. Focus the eye- piece of the telescope on the cross-wires or needle-point. Turn the telescope to some very distant object, and focus the object-glass by the parallel method described on p. 369. Turn the telescope to look into the collimator ; illuminate the slit, and then focus it by altering its position with reference to the lens of the collimator. When the slit is in focus, the light issuing from the collimator forms a series of pencils of parallel rays.^ * This is a very important adjustment ; if it be properly carried out the direction of the rays forming an image after reflexion or refraction at the surface of a prism, and hence the circle readings, will be the same, no matter to what extent the prism may be moved parallel to itself about the spectrometer table. In the absence of such an adjustment the measurements would require a piism with in- definitely small width of face and its edge coincideru with the axis o( rotation. It will be seen that the faces of a prism for accurate optical work must be plane, A prism which shews by the alteration of focus which it produces that its faces are not plane must be discarded except for roughly approximate measurements. Ch. XIV. § 62.] Spectra, Refractive Indices, Grc. 387 In experiments in which a prism is used it is generally necessary that the edge of the prism should be parallel to the axis of rotation of the telescope. Turn the telescope to view the slit directly. Fix by means of soft wax a hair or silk fibre across the slit, so that it may appear to coincide with the horizontal cross- wire or point of the needle when seen through the instrument; or, as is often more con- venient, cover up part of the slit, making the junction of the covered and uncovered portions coincide with the horizontal wire. Fix the prism with wax or cement on to the levelling table in the centre of the instrument, so that the light from the collimator is reflected from two of its foces, and adjust it by hand, so that the two reflected images of the slit can be brought in turn into the field of view of the telescope. Alter the set screws of the levelling table until the image of the hair across the slit when reflected from either of the two faces, and seen through the tele- scope, coincides with ' the intersection of the cross-wires. When this is the case the prism is in the required positioa The edge of the prism may also be adjusted to be parallel to the axis of rotation by setting the two faces suc- cessively at right angles to the line of collimation of the telescope. This may be done with great accuracy by the following optical method. Illuminate the cross- wires of the telescope, and adjust the face of the prism so that a reflected image of the cross-wires is seen in the field of view of the telescope coincident with the wires themselves. This can only be the case when the pencil of light from the inter- section of the wires is rendered parallel by refraction at the object-glass of the telescope, and reflected normally by the face of the prism, so that each ray retiums along its own path {^see% p). An aperture is provided in tfie eye- piece tubes of some instruments for die purpose of illuminating the wires ; in the absence of any such provision, a piece -' plane glass, placed at a suitable angle in front of th piece, may be used. It is sometimes difficult to catc cc 388 Practical Physics. [Ch. XIV. § 6x of the reflected image in the first instance, and it is generally advisable, in consequence, to make a rough adjustment idth the eye-piece removed, using a lens of low magnifying power instead. When fixing the prism on to the table, it is best to take care that one face of the prism is perpendicular to the line joining two of the set screws of the levelling table. Level this face first The second face can then be adjusted by simply altering the third screw, which will not disturb the first face. It is well to place the prism so that the light used passes as nearly as possible through the central portion of the object-glasses of die collimator and telescope. Measurements with the Spectrometer, (i) To verify the Law of Reflexion. This requires the table on which the prism is fixed to be capable of motion round the same axis as the telescope, and to have a vernier attached. Adjust the apparatus so that the reflected image of the slit coincides with the cross- wire, and read the position of the telescbpe and prism. The slit should be .made as narrow as possible. If the instrument has two verniers for the telescope opposite to each other, read both and take the mean of the readings. Errors of centering are thus eliminated. Move the prism to another position, adjust the telescope as before, and take readings of the position of the prism and telescope. Subtract these results firom the former re- spectively. It will be found that the angle moved through by the telescope is always twice that moved through by the prism. (a) To Measure the Angle of a Prism, (a) Keeping the prism fixed — Adjust the prism so that an image of the slit can be seen distinctly by reflexion firom Ch. XIV. § 6a.] Spectra^ Refractive Indices^ Src 389 each of two of its faces, and its edge is parallel to the axis of rotation of the telescope. Adjust the telescope so that the image of the slit re- flected from one face coincides with the vertical cross-wire, and read the verniers. Move the telescope until the same coincidence is observed for the image reflected from the second face, and read again. The difference of the two readings is twice the angle required, provided the incident light is parallel. ip) Keeping the telescope fixed, — Move the prism until the image of the slit reflected from one face coincides with the vertical cross-wire, and read the verniers for the prism. Turn the prism until the same coincidence is observed for the other face, and read again. Then the defect of the difference of the two readings from 180^ is the angle required. Verify by repeating the measiu-ements. EtXpefitnents* (i) Verify the law of reflexion. (2) Measure by methods (a) and (.<] Oie angle of the given prism. Enter results thus: — (i) Displacement of telescope . 5** 42' 24^0' 15' (2) Angle of prism — „ prism . 2® 51' 12'' o' o" By method (a) 60"* 7' 30^' 60** 7' 50" mean 60** 7' 40" By method (d) 60** 8' 15'' 60^ 7' 45" mean 60** 8' o" (3) To Measure the Refractive Index of a Prism. First Method, — ^The spectrometer requires adjusting and the prism levelling on its stand, as before. The angle of the prism must be measured, as described. To obtain an accurate result, it is necessary that the light which falls on the face of the 390 Practical Physics. [Ch. XIV. § 6s. prism should be a parallel pencil One method of obtaining this has already been given. The following, due to Professor Schuster, may often be more convenient, and is, moreover, more accurate. Let us suppose that the slit is illuminated with homogeneous light, a sodium flame, for example, and the prism so placed that the light passes through it, being deflected, of course, towards the thick part. Place the tele- scope so as to view the refracted image. Then it will be found that, on turning the prism round continuously in one direction, the image seen appears to move towards the direction of the incident light, and after turning through some distance the image begins to move back in the oppo- site direction and again comes into the centre of the field. There are thus, in general, for a given position of the tele- scope, two positions of the prism, for which the image can be brought into the centre of the field of the telescope. In one of these the angle of incidence is greater than that for minimum deviation, in the other less. Turn the prism into the first of these positions ; in general the image will appear blurred and indistinct Focus the telescope until it is dear. Then turn the prism into the second position. The image now seen will not be clear and in focus unless the colli- mator happens to be in adjustment Focus the collimator. Turn the prism back again into the first position and focus the telescope, then again to the second and focus the colli* mator. After this has been done two or three times, the slit will be in focus without alteration in both positions of the prism, and when this is the case the rSys which fall on the telescope are parallel ; for since the slit remains in focus, its virtual image formed by the prism is at the same distance from the telescope in the two positions of the prism ; that is to say, the distance between the prism and the virtual image of the slit is not altered by altering the angle of incidence, but, since the image corresponds to the primary focal line {^su § q), this can only be the case when that distance is infinite — that is, when the rays are parallel on leaving the Ch, XIV, § 62.] Spectra^ Refractive Indices^ &c. 391 prism ; and since the faces of the prism are plane, the rays emerging from the collimator are parallel also. Thus both telescope and collimator may be brought into adjustment. The simplest method of measuring the refractive index IS to observe the angle of the prism and the minimum de- viation. We have seen how to measure the former. For the latter, turn the telescope to view the light coming directly from the collimator. When the prism is in position, it of course intercepts the light, but it can generally be turned round so as to allow sufficient light for the purpose to pass on one side of it Clamp the telescope and adjust with the tangent screw until the intersection of the cross-wires or the end of the needle comes exactly into the centre of the slit ; then read the scale and vernier. Repeat the observation several times and take the mean of the readings. If it be impossible to turn the prism without removing it from its place, so as to view the direct image, a method to be de- scribed later on may be used. Turn the prism to receive on one face the light emerging from the collimator, and move the telescope to view the refracted image. Place the prism so that the deviation of the refracted light is a minimum. To determine this position accurately, tiim the prism round the axis of the circle so that the refracted image appears to move towards the direction of the incident light, and continue the motion until the image appears to stop. This position can easily be found roughly. Bring the cross-wire of the telescope to cover the image of the slit, and again turn the prism slightly first one way and then the other. If for motion in both directions the image appears to move away from the direction of the incident light, the prism is in the required position. In general however, for the one direction of rotation the motion of the image will be towards the direct light, and the prism must be turned until the image ceases to move in that direction. The first setting gave us an approximate position for the i9^ Practical Physics. (Ch. XIV. § 6a. prism. By bringing the cross-wires over the image, and then moving the prism, we are able to detect with great ease any small motion which we should not have noticed had there been no mark to which to refer it Having set the prism, place the telescope, using the clamp and tangent screw so that the cross- wire bisects the image of the slit, and read the vernier. Displace the prism and telescope, set it again, and take a second reading. Repeat several times. The mean of the readings obtained will be the minimum deviation reading, and the difference between it and the mean of the direct readings the minimum deviation. With a good instrument and reasonable care the readings should not differ among themselves by more than i'. Having obtained the minimum deviation d, and the angle of the prism /, the refractive index /a is given by sin|(D+i) sm4/ To check the result, the prism should be turned so that the other face becomes the face of incidence, and the devia- tion measured in the opposite direction. If we cannot observe the direct light, we may note the deviation reading on each side of it — that is, when first one face and then the other is made the face of incidence — the difference between the two readings is twice the minimum deviation required, while half their sum gives the direct reading. To determine the refractive index of a liquid we must enclose it in a hollow prism, the faces of which are pieces of accurately worked plane parallel glass, and measure its refractive index in the same way as for a solid. Experiment. — Determine the refractive index of the given prisxB. Cm. XIV. S 63.1 Spectra, Refractive Indices, &c 393 Enter results thus: — Direct nading Deviatioo reading (*) Deviation reading (a) 183** 15' 4<y' 143" 29' 223** 2' iSy, 15' 5</' 143'' 28' 50 223** 1' 30" 183** 15' 30^' 143° 29' ic/' 223*' 1' 30" Mean 183** 15' 4c/' 143*^ 29^ 223*^ 1' 40" Deviation (i) . . . • 39^ 46' 40" Deviation (2) . . . . 39** 46' o" Mean ••••39'' 46' 20^ Angle of the prism • . . (xP ot of* Hence /i= 1*5295. Second Method. — ^The following is another method of measuring the refractive index, which is useful if the angle of the prism be sufficiently small. Let the light from the collimator faU perpendicularly on the face of incidence. Then if 1 be the angle of the prism and d the deviation, since, using the ordinary notation, <^ = <^'ao; A f = | ^ = D+^ and /A = sin ^/sin ^' = sin (D+O/^in L We require to place the prism so that the face of incidence is at right angles to the incident light Turn the telescope to view the direct light and read the vernier. Place the prism in position and level it, as ahready described Turn the telescope so that the vernier reading differs by 90^ from the direct reading. Thus, if the direct reading be 183® 15' 30", turn the telescope till the vernier reads 273** 15' 30". This can easily be done by the help of the clamp and tangent screw. Clamp the telescope in this position ; the axes of the collimator and telescope are now at right angles. Turn the prism until the image of the slit reflected from one Cace comes into the field, and adjust it until there is 394 Practical Physics. [Ch. XIV. § 6x coincidence between this image and the cross-wire. The light falling on the prism is turned through a right angle by the reflexion. The angle of incidence is therefore 45 • exactly. Read the vernier attached to the table on which the prism rests, and then turn the prism through 45** exactly, so as to decrease the angle of incidence ; then the iajc^ of incidence will evidently be at right angles to the incident light Now turn the telescope to view the refracted image, and read the vernier; the difference between the reading and the direct reading is the deviatioa The angle of the prism can be measured by either of the methods already described ; it must be less than sin "' (i //i), which for glass is about 42®, otherwise the light will not emerge from the second face, but be totally reflected there. The refractive index can now be calculated from the formula. A similar observation will give us the angle of incidence at which the light falls on any reflecting surface ; thus turn the telescope to view the direct light, and let the vernier reading be a, then turn it to view the reflected image, and let the reading be p. Then a—/? measures the deflection of the light, and if ^ be the angle of incidence, we can shew that the deviation is 180**— 2^. .% 180**— 2<^ = o— ^; Experiment, — Determine the refractive index of the given prism for sodium light. Enter the results thus:— Angle of prism . 15** 35' 20'' Direct reading Deviation reading iZf 15' 10" 191" 53' 30" 183^ 15' 50" 191** 54' 20"' 183° 15' 30^' 191° 53' 40" Mean 183° 15' 30''' 191° 53' 50'' Deviation . • 8° 38' 20^ Value of /A . i*S27i. Ch. XIV. § 62.] Spectra^ Refractive Indices^ &c, 395 (4) To Measure the Wave- Length of Light by means of a Diffraction Grating. A diffraction grating consists of a number of fine lines ruled at equal distances apart on a plate of glass— a trans- mission grating ; or of speculum metal — ^a reflexion grating. We will consider the former. If a parallel pencil of homo- geneous light fall normally on such a grating, the origin of light being a slit parallel to the lines of the grating, a series of diffracted images of the slit will be seen, and if 0^ be the deviation of the light which forms the n\h image, reckoning from the direction of the incident light, d the distance between the centres of two consecutive lines of the grating, and X the wave-length, we have ^ X = -</sin 0^ n The quantity d is generally taken as known, being determined at the time of ruling the grating. The spectro- meter is used to determine 6^. The telescope and collimator are adjusted for parallel rays, and the grating placed on the table of the instrument with its lines approximately parallel to the slit For con- venience of adjustment it is best to place it so that its plane is at right angles to the line joining two of the levelling screws. The grating must now be levelled, i.e. adjusted so that its plane is at right angles to the table of the spectro- meter. This is done by the method described above for the prism. Then place it with its plane approximately at right angles to the incident light, and examine the diffracted images of the slit The plane of the grating is at right angles to the line joining two of the levelling screws ; the third screw then can be adjusted without altering the angle between the plane of the grating and the table of the spectrometer. Adjust the third screw until the slit appears as distinct as possible ; the lines of the grating will then be parallel to the slit. > See Glazebrook, Physical Optics,^ 189. 396 Practical Physics, iCh. XIV, 5 6x Turn the table carrying the grating so as to allow the direct light to pass it ; adjust the telescope so that the vertical cross wire bisects the image of the slit seen directly, and read the vernier. This gives us the direct reading. Place the grating with its plane accurately perpendicular to the incident rays, as described above (p. 393), and turn the telescope to view the diffracted images in turn, taking the omrespond- ing readings of the vernier. The difference between these and the direct reading gives us the deviations ^|, 6^ &c A series of diffracted images will be formed on each side of the direct rays. Turn the telescope to view the second series, and we get another set of values of the deviation ^], ^21 &C. If we had made all our adjustments and observations with absolute accuracy, the corresponding values ^1, ^1, &c., would have been the same ; as it is their mean will be more accurate than either. Take the mean and substitute in the formula X = -^sin0^ n We thus obtain a set of values of X. If the light be not homogeneous, we get, instead of the separate images of the slit, more or less continuous spec- tra, crossed it may be, as in the case of the solar spectrum, by dark lines, or consisting, if the incandescent body be gas at a low pressure, of a series of bright lines. In some cases it is most convenient to place the grating so that the light falls on it at a known angle, ^ say. Let ^ be the angle which the diffracted beam makes with the normal to the grating, and B the deviation for the irth image, ^ and ^ being measured on the same side of the normal, then it may be shewn that ftnd « X = </(sin ^+sin ^^) = d{&\vi <& + sin {fi — <^)) . Ch. XIV. § 62]. Spectra^ Refractive Indices^ &c. 397 The case of greatest practical importance is when the deviation is a minimum, and then ^ = ^ = ^ ^, so that if 6^ denote the minimum deviation for the nih diffracted image, we have n In the case of a reflexion grating, if ^ and ^ denote the angles between the normal and the incident and reflected rays respectively, ^ and ^ now being measured on opposite sides of the normal, the formula becomes n\s=id (sin ^— sin <^) ; and if B be the deviation If the value of d be unknown, it may be possible to find it with a microscope of high power and a micrometer eye- piece. A better method is to use the grating to measure 0^ for light of a known wave-length. Then in the formula, ^Xs^sin^M we know A, n^ and 6^^ and can therefore determine d. Experiment. — Determine by means of the given grating the wave-length of the given homogeneous light Value </« 1 a Paris inch 3000 B -0009023 cm. rah les of deviations, each the mean of three observations- A 2. 3 3* 44' 30^' t 29' o'' 3^ 44' 45" t 29' 45" 11° \f 30" Mean 3** 44' 37"-5 7** 29^ 22''*5 ,,0 jy/ Y''^ Values of X • • • Tenth metres ' " 5895 5893 Mean • • 5915 5901 ^ A tenth metre * is 1 metre divided by io"V 398 Practical Physics, [Ch. XIV, f ^3- 63. The Optioal Bench. The optical bench (fig. 39) consists essentially of a graduated bar carrying three upright pieces, which can slide along the bar ; the second upright from the right in the Fio. 30. figure is an addition to be described later. The uprights are provided with verniers, so that their positions relatively to the bar can be read. To these uprights are attached metal jaws capable of various adjustments ; those on the first and second uprights can rotate about a vertical axis through its centre and also about a horizontal aids at right angles to the upright ; they can also be raised and lowered. The second upright is also capable of a transverse motion at right angles to the length of the bar, and the amount of this motion can be read by means of a scale and vernier. The jaws of the first upright generally carry a slit, those of the second are used to hold a bi-prism or apparatus re- quired to form the diffraction bands. To the third upright is attached a Ramsden's eye-piece in front of which is a vertical cross- wire ; and the eye-piece and cross-wire can be moved together across the field by means of a micrometer screw. There is a scale attached to the fiame above the eye-piece, by which the amount of displacement can be measured. The whole turns of the screw are read on the scale by means of a pointer attached Ch. XIV. $ 63.] Spectra^ Refractive Indices^ &c. 399 to the eye-piece. The fractions of a turn are given by the graduations of the micrometer head. The divisions of the scale are half-millimetres and the micrometer head is divided into 100 parts. {i) To Measure the Wave-Length of Light by means of FresnePs Bi prism. The following adjustments are required : — (i) The centre of the slit, the centre of the bi-prism, and the centre of the eye-piece should be in one straight line. (2) This line should be parallel to the graduated scale of the bench. (3) The plane face of the bi-prism should be at right angles to this line. (4) The plane of motion of the eye-piece should also be at right angles to the same line. (5) The cross-wire in the eye-piece, the edge of the prism, and the slit should be parallel to each other, and vertical, that is to say, at right angles to the direction of motion of the eye-piece. To describe the adjustments, we shall begin with (5). Focus the eye-piece on the cross- whre, and by means of the flat disc to which it is attached, turn the latter round the axis of the eye-piece until it appears to be vertical; in practice the eye is a sufficiently accurate judge of when this is the case. Draw the third upright some way back, and insert between it and the slit a convex lens.* Illuminate the slit by means of a lamp, and move the lens until a real image of the slit is formed in the plane of the cross- wire. Turn the slit round by means of the tangent screw until this image is parallel to the crossrwire. The slit must be held securely and without shake in the jaws. Move the eye-piece up to the slit an(} adjust the vertical and micrometer screws until the axis of the eye-piece appears to pass nearly through the centre of the slit, turning at the same time the eye-piece round the vertical axis until its axis appears parallel to the scale. This secures (4) approximately. * This is shewn in the figure. 400 Practical Physics, (Ch. XIV. \ 63. Draw the eye-piece away from the slit, say 20 or 30 cm. off, and place the bi-prism in position, turning it about until its plane face appears to be at right angles to the scale of the bench. This secures (3) approximately. Look through the eye- piece. A blurred image of Fresnel's bands may probably be visible. By means of the traversiiig screw move the second upright at right angles to the scale until this image occupies the centre of the field. If the bands be not visible, continue to move the screw until they come into the field. It may be necessary to alter the height of the bi-piism by means of the vertical adjustment so that its centre may be at about the same level as those of the slit and eye-piece. By means of the tangent screw turn the bi-prism round the horizontal axis at right angles to its own plane until the lines appear bright and sharp. Adjustment (5) is then complete. Now draw the eye-piece back along the scale ; if the lines still remain in die centre of the field of view, it follows that the slit, the centre of the bi-prism, and the centre of the eye-piece are in one straight line parallel to the scale. If this be not the case, alter the position of the eye-piece by means of the micrometer screw and that of the bi-prism by means of the traversing screw with which the second stand is furnished, until the lines are seen in the centre of the field for all positions of the eye-piece along the scale bar of the instrument Adjustments (i) and (2) have thus been effected. For (3) and (4) it is generally sufficient to adjust by eye, as already described. If greater accuracy be required, the following method will secure it Move the lamp to one side of the slit and arrange a small mirror so as to reflect the light through the slit and along the axis of the instrument. The mirror must only cover one-half of the slit, which will have to be opened some- what widely. Place your eye so as to look through the other half of the slit in the same direction ^ the light Images Ch. XIV. §63.) Spectra^ Refractive Indices^ &c. 401 of the slit reflected from the feces of the bi-prism and probably from other parts of the apparatus will be seen. Suppose the flat face of the bi-prism is towards the slit Turn the prism round a vertical axis until the image reflected at the flat face appears directly behind the centre-line of the bi-prism, then clearly the plane of the bi-prism is at right angles to the incident light, and that is parallel to the scale. In making the adjustment, the stand holding the prism should be placed as far as may be from the slit If the bevelled face be towards the slit, two images will be seen, and these must be adjusted symmetrically one on each side of the centre. To adjust the eye-piece employ the same method, using the image reflected from the front lens or from one of the brass plates which are parallel to it To do this it may be necessary to remove the bi-prism — if this be the case, the eye-piece adjustment must be made first. As soon as the adjustments are made the various moving pieces must be clamped securely. It is necessary for many purposes to know the distance between the slit and the cross- wire or focal plane of the eye- piece. The graduations along the bar of the instrument will not give us this directly; for we require, in addition, the horizontal distance between the zero of the vernier and the slit or cross-wire respectively. To allow for these, take a rod of known length, a centimetres suppose ; place one end in contact with the slit, and bring up the eye-piece stand until the other end is in the focal plane. Read the distance as given by the scale between the slit and eye-piece uprights ; let it be b centi- metres. Then clearly the correction a—b centimetres must be added to any scale reading to give the distance between the slit and the eye-piece. This correction should be de- termined before the bi-prism is finally placed in position. To use the bi-prism to measure A, the wave-length of D D 402 Practical Physics. (Cn. XIV. § 63. light, we require to know r, the distance between the virtua) images formed by the bi-prism, x the distance between con secutive bright bands, and a the distance between slit and eye-piece.' Then we have X = ^* a The distance x is measured by means of the micrometer attached to the eye-piece. In order that x may be large, c should be small and a large. This is attained by making the distance between the slit and the prism small, 10 to 20 cm., and that between the prism and the eye-piece considerable. Of course the bands are fainter and less distinct if this distance be very large; it most therefore not exceed a certain limit, which depends greatly on the source of light used Suppose we have a Bunsen burner with a sodium bead in it In making the measurement of x^ the micrometer screw of the eye- piece should be always turned in the same direc- tion. This avoids the error of * lost time ' due to any shake in the screw or looseness between the screw and the nut. Turn the screw to carry the cross- wire as near to one edge of the field as is convenient and set it on the centre of a bright band. Read the scale and micrometer; let the reading be 10*35. Turn the screw until the wire is over the next bright band and read again; let the reading be 1072. Proceed thus across the field, reading the position of every bright line, and taking an even number, say ten or twelve readings. Let them be (i) 1035 (6) 12-15 (2) 1072 (7) 12-53 (3) 1 1 07 (8) 12-88 (4) 11-45 (9) 1324 (5) 1 1 -Si (10) 13-59 • Sec Glazcbrook, Physical Optks^ dup, v. Ch. XIV. § 63.] Spectra^ Refractive Indices^ &c 403 Subtract the first from the sixth, the second from the seventh, and so on. Then (6)-(i)=r8o (7)-(2)=r8i (8)-(3)=i-8i (9)-(4)=i79 (io)-(5)=i78 Mean . . • 1798 £ach of these differences is the space coveted by a group of six bright lines. Take the mean. We have 1798. Dividing by five we get the mean value for x. Thus X == '359 mm, To determine a we have only to read the verniers at the slit and eye-piece respectively, take the difference and correct it as already described for index error. To determine r, draw the eye-piece away to about 50 centimetres from the slit and insert between the prism and the eye-piece a convex lens. It is convenient to have a fourth sliding upright arranged to carry this, as is shewn in the figure. Two positions for this lens can in general be found, in each of which it will form in the focal plane of the eye- piece distinct images of the two virtual images of the slit. The distance between these two images in each of these two positions respectively can be found by means of the micrometer screw. Let them be Cx and c^ then it is easy * to shew that c == Vcx c^ ' We may replace the bi-prism by Fresnel's original apparatus of two mirrors, arranging the bench so as to give the frindamental interference experiment Or, again, instead of two mirrors, we may obtain in^ terference between the light coming from the slit and its > See Glazebrook, Physical Optics^ p. Tf8. D D2 404 Practical Pf^sics. [Ch. xrv. % 6> image by reflexion at a large angle of incidence from a plane glass surface (Lloyd's Experiment)* Diffraction Experiments. The apparatus may be used to examine the efifects of diffraction by various forms of aperture. The plate with the aperture is placed in the second up- right in the place of the bi-prism. If we have a single edge at a distance a from the silt, and if b be the distance between the edge and the eye-piece, X the distance between two bright lines Then » '=v/( Ma±^^ If the obstacle be a fibre of breadth ^, then * = £-, where b is distance between the fibre and the screen or eye-piece. This formula, with a knowledge of the wave-length of the light, may be used to measiure the breadth of the fibre (Young's Eriometer.) In order to obtain satisfactory results from diffraction experiments a very bright beam of light is required. It b best to use sunlight if possible, keeping the beam directed upon the slit of the optical bench by means of a heliostat. Experiments. — Measure the wave-length of light by means of t lie bi-prism. Enter results thus: — a " 56 cm. *•« x>359 cm., (mean of 5) ^-•092 cm., ( „ 3) X =» '0000589 cm. * Glazebrook's Physical Optics^ P* I72. \ i ' ■ -> ■ I \ 40s CHAPTER XV. POLARISED LIGHT, C F On the Determination of the Position of the Plane of Polarisation} The most important experiments to be made with polarised light consist in determining the position of the plane of polarisation, or in measuring the angle through which that plane has been turned by the passage of the light through a column of active substance, such as a solution of sugar, turpentine, or various essential oils, or a piece of quartz. The simplest method of making this measurement is by the use of a Nicol's or other polarising prism. This is mounted in a cylindrical tube which is capable of rotation about its own axis. A graduated circle is fixed with its centre in the axis of the tube, and its plane at right angles to the axis, and a vernier is attached to the tube and rotates with it, so that the position, with reference to the circle, of a fiducial mark on the tube can be found. In some cases the vernier is fixed and the circle turns with the Nicol. If we require to find the position of the plane of polarisation of the incident light, we must, of course, know the position of the principal plane of the Nicol relatively to the circle. If we only wish to measure a rota- tion a knowledge of the position of this plane is unnecessary, for the angle turned through by the Nicol is, if our adjust- ments be right, the angle turned through by the plane of polarisation. For accurate work two adjustments are necessary : — (i) All the rays which pass through the Nicol should be parallel. (2) The axis of rotation of the Nicol should be paralle* to the incident light To secure the first, the source of light should be small; > See Glazebrook, Phjfsual Optics^ chap. xiv. 4o6 Practical Physics. [Ch. xv. { 6> in many cases a brightly illuminated slit is the best It should be placed at tb.c principal focus of a convex lens ; the beam emerging from the lens will then consist of parallel rays. To make the second adjustment we may generally consider the plane ends of the tube which holds the Nicol as perpendicular to the axis of rotation. Place a plate of glass against one of these ends and secure it in this position with soft wax or cement The incident beam falling on this plate is reflected by it Place the plate so that this beam after reflexion retraces its path. This is not a difficult matter ; i^ however, special accuracy is required, cover the lens from which the rays emerge with a piece of paper with a small hole in it, placing the hole as nearly as may be over the centre of the lens. The light coming through the hole is reflected by the plate, and a spot of light is seen on the paper. Turn the Nicol about until this spot coincides with the hole ; then the incident light is evi- dently normal to the plate —that is, it is parallel to the axis of rotation of the Nicol. If still greater accuracy be required, the plate of glass may be dispensed with, and a reflexion obtained from the front face of the Nicol. This, of course, is not usually normal to the axis, and hence the reflected spot will never coincide with the hole, but as the Nicol is turned, it will describe a curve on the screen through which the hole is pierced If the axis of rotation have its proper position and be parallel to the direction of the incident light, this curve will be a drde with the hole as centre. The Nicol then must be adjusted until the locus of the spot is a circle with the hole as centre When these adjustments are completed, if the incident light be plane-polarised, and the Nicol turned until there is no emergent beam, the plane of polarisation is parallel to the principal plane of the Nicol; and if the plane of polarisation be rotated and the Nicol turned again till the emergent beam is quenched, the angle turned through by Ch.XV. §63.] Polarised Light, 407 the Nicol measures the angle through which the plane of polarisation has been rotated. But it is difficult to determine with accuracy the position of the Nicol for which the emergent beam is quenched. Even when the sun is used as a source of light, if the Nicol be placed in what appears to be the position of total extinction, it may be turned through a considerable angle without causing the light to reappear. The best results are obtained by using a very bright narrow line of light as the source — the filament of an incandescence lamp has been successfully employed by Mr. McConnel — as the Nicol is turned, a shadow will be seen to move across this line from one end to the other, and the darkest portion of the shadow can be brought with considerable accuracy across the centre of the bright line. Still, for many pur- poses, white light cannot be used, and it is not easy to secure a homogeneous light of sufficient brightness. Two principal methods have been devised to overcome the difficulty ; the one depends on the rotational properties of a plate of quartz cut normally to its axis ; the other, on the fact that it is comparatively easy to determine when two objects placed side by side are equally illuminated if the illu- mination be only faint. We proceed to describe the two methods. 64. The Bi-quartz. If a plane-polarised beam of white light fall on a plate of quartz cut at right angles to its axis, it has, as we have said, its plane of polarisation rotated by the quartz. But, in addition to this, it is found that the rays of different wave- lengths have their planes of polarisation rotated through different angles. The rotation varies approximately inversely as the square of the wave-length; and hence, if the quartz be viewed through another Nicol's prism, the proportion ot light which can traverse this second Nicol in any position will be different for different colours, and the quartz will appear coloured. Moreover, the colour will vary as the 4o8 Practical Physics, [Ch. XV. $64. analysing Nicol, through which the quartz is viewed, is turned round. If the quartz be ab(^ut 3*3 mm. in thickness, for one position of the Nicol it Mrill appear of a peculiar neutral grey tint, known as the tint of passage. A slight rotation in one direction will make it red, in the other blue. After a little practice it is easier to determine, even by eye, when this tint appears, than to feel certain when the light is completely quenched by a NicoL It can be readily shewn moreover that when the quartz gives the tint of passage, the most luminous rays, those near the Fraunhofer line e, are wanting from the emergent beam ; and if the quartz have the thick- ness already mentioned, the plane of polarisation of these rays has been turned through 90**. A still more accurate method of making the observation is afforded by the use of a bi-quartz. Some specimens of quartz produce a right-handed, others a left-handed rotation of the plane of polarisation of light traversing thencL A bi- quartz consists of two semicircular plates of quartz placed so as to have a common diameter. The one is right- handed, the other left The two plates are of the same thickness, and therefore produce the same rotation, though in opposite directions, in any given ray. If, then, plane- polarised white light pass normally through the bi-quartz, the rays of different refrangibilities are differently rotated, and that too in opposite directions by the two halves, and if the emergent light be analysed by a Nicol, the two halves will appear differently coloured If, however, we place the analysing Nicol so as to quench in each half of the bi -quartz the ray whose plane of polarisation is turned through 90** — that is to say, with its principal plane parallel to that of the polariser — light of the same wave-length will be absent ftom both halves of the field, and the other rays will be present in the same proportions in the two; and if the thickness of the bi-quartz be about 3-3 mm this common tint will be the tint of passage. A very slight rotation of the analyser in one direction renders one half red. the other blue, while if Ch. XV. § 64.J Polarised Light 409 the direction of rotation be reversed, the first half becomes blue, the second red Hence the position of the plane of polarisation of the ray which is rotated by the bi-quartz through a certain definite angle can be very accurately de- termined. A still better plan is to form the light after passing the analyser into a spectrum. If this be done in sudi a way as to keep the rays coming from the two halves of the bi-quartz distinct — e.g. by placing a lens between the bi-quartz and the slit and adjusting it to form a real image of the bi-quartz on the slit, while at the same time the slit is perpendicular to the line of separation of the two halves — two spectra will be seen, each crossed by a dark absorption band. As the analysing Nicol is rotated the bands move in opposite directions across the spectrum, and can be brought into coincidence one above die other. This can be done with great accuracy and forms a very delicate method. Or we may adopt another plan with the spectroscope : we may use a single piece of quartz and form the light which has passed through it into a spectrum, which will then be crossed by a dark band ; this can be set to coincide with any part of the spectrum. This is best done by placing the telescope so that the cross-wire or needle-point may coincide with the part in question, and then moving the band, by turning the analyser, until its centre is under the cross-wire. Fig. 40b E3-^^f- K Fig. 40 gives the arrangement of the apparatus : l is the lamp, a the slit, and c the collimating lens. The parallel rays fall on the polarising Nicol n and the bi- quartz B. They then traverse the tube t containing the active rotatory substance and the analysing Nicol n', foiling 4IO Practical Physics. [Ch. XV. §64. on the lens m which forms an image of the bi-quartz on the slit s of the small direct-vision spectroscope. If we wish to do without the spectroscope, we can remove both it and the lens m and view the bi-quartz either with the naked eye or with a lens or small telescope adjusted to see it distinctly. If we use the single quartz, we can substitute it for the bi-quartz, and focus the eye-piece of the telescope to see the first slit a distinctly, and thus observe the tint of passage. The quartz plate may be put in both cases at either end of the tube t. If it be placed as in the figure, and the apparatus is to be used to measure the rotation produced by some active substance, the tube should in the first in- stance be filled with water, for this will prevent the neces- sity of any great alteration in the adjustment of the lens u or in the focussing of the telescope, if the lens be not used, between the two parts of the experiment The mode of adjusting the Nicols has been already described. The light should traverse the quartz parallel to its axis, and this should be at right angles to its faces. This last adjustment can be made by the same method as was used for placing the axis of the Nicol in the right position, pro- vided the maker has cut the quartz correctly. In practice it is most convenient to adjust the quartz by hand, until the bands formed are as sharp and clear as may be. Care must be taken that each separate piece of the apparatus is securely fastened down to the table to prevent any shake or accidental disturbance. If a lens is used at m, it is best to have it secured to the tube which carries the analysing Nicol, its centre being on the axis of this tube; by this means it is fixed relatively to the Nicol, and the light always comes through the same part of the lens. This is important, for almost all lenses exert a slight depolarising effect on light, which differs ap- preciably in different parts of the lens. For most purposes Ch. XV. § 64. J Polarised Light 411 this is not very material, so long as we can be sure that the effect remains the same throughout our observations. This assurance is given us, provided that the properties of the lens are not altered by variations of temperature, if the lens be fixed with reference to the principal plane of the analyser, so that both lens and analyser rotate together about a common axis. « One other point remains to be noticed. If equality of tint be established in any position, and the analyser be then turned through 180**, then, if the adjustments be perfect, there will still be equality of tint. To ensure accuracy we should take the readings of the analysing Nicol in both these positions. The difference between the two will pro- bably not be exactly 180** ; this arises mainly from the fact that the axis of rotation is not accurately parallel to the light. The mean of the two mean readings will give a result nearly free from the error, supposing it to be small, which would otherwise arise from this cause. To attain accuracy in experiments of this kind needs considerable practice. Experiments, (i) Set up the apparatus and measure the rotation produced by the given plate of quartz. (2} Make solutions of sugar of various strengths, and verify the law that the rotation for light of given wave-length varies as the quantity of sugar in a unit of volume of the solution. Enter results thus: — Thickness of quartz : — I'oi cm. 1*012 cm. 1*011 cm. Mean I'oii cnL Analyser readings without quartz plate. Position A Position B 6^ f 186° la 6»9 186** 12' 6** 8' 186** 9' 6° 6' 186° II' Me^n 6° f 30" Mean 186° 10' 30" Mean of the two • . . . 96^9' 412 Practical Physics. (Ch. XV. § 64. Analyser readings with quartz plate. Position A Podtioa b 280** 4/ 360 + lOO* 48' 280** 45' + lOO* 4/ 280^ 46' 100** 49' 280** 48' 100** so' Mean 280° 46' 30" Mean 360+100** 48' 30' Mean of the two 370** 47' 30 Mean rotation 274° 38 30 Rotation deduced from position A . 274® 39' o'' H B . 274^ 38' o" 65. Shadow Polarimeten. The theory of these, as has been stated, all turns on the fact that it is comparatively easy to determine when two objects placed side by side are equally illuminated, the illumination being faint Suppose, then, we view through a small telescope or eye-piece placed behind the analyser a circular hole divided into two parts across a diameter, and arranged in such a way that the planes of polarisation of the light emerging from the two halves are inclined to each other at a small angle. For one position of the analyser one half of the field will be black, for another, not very different, the other half will be black, and for an intermediate position the two halves will have the same intensity. The analyser can be placed with the greatest nicety in the position to produce this. If now the planes of polarisation of the light from the two halves of the field be each rotated through any the same angle and the analyser turned until equality of shade is re-established, the angle through which the analyser turns measiures the angle through which the plane of polari- sation has been rotated. Whatever method of producing the half-shadow field be adopted, the arrangement of apparatus will 4>e similar to hat shewn in fig. 40, only b will be the half-shadow plate^ CiL XV. § esl Polarised Light. 413 and instead of the lens m and the spectroscope s we shall have a small telescope adjusted to view the plate b. In nearly all cases homogeneous light must be used for accurate work. Excellent results can be obtained by placing a bead of sodium on a small spoon of platinum gauze just inside the cone of a Bunsen burner, and then allowing a jet of oxygen to play on the gauze. Lord Rayleigh has found that a good yellow light is given by passing the gas supplied to a Bunsen burner through a small cylinder containing a finely divided salt of sodium, keeping the cylinder at the same time in a state of agitation, while Dr. Perkin passes the gas over metallic sodium in an iron tube which is kept heated. The brilliancy of the light is much increased by mixing oxygen with the coal gas as in the oxyhydrogen light Whenever a sodium flame is used, it is necessary that the light should pass through a thin plate of bichromate of potassium, or through a small glass cell containing a dilute solution of the same salt, to get rid of the blue rays from the gas. In almost all cases the half-shadow arrangement may be attached to either the polariser or the analyser. If the latter plan be adopted, it must, of course, turn with the analyser, and this is often inconvenient ; the other arrange- ment, as shewn in fig. 40, labours under the disadvantage that the telescope requires readjusting when the tube with the rotating liquid is introduced. We will mention briefly the various arrangements which have been suggested^ for producing a half-shadow field, premising, however, that as the sensitiveness depends both on the brightness of the light and the angle between the planes of polarisation in the two halves of the field, it is convenient to have some means of adjusting the latter. With a bright light this angle may conveniently be about 2^ It is also important that the line of separation between * See also Glazebrook, Physical Optics^ chap, ziv. 414 Practical Physics. [Ch. XV. § 65. the two halves should be very narrow, and sharp, and distinct (i) Jellett's prism: — The ends of a long rhomb of spar are cut off at right angles to its length, and then the spar cut in two by a plane parallel to its length and inclined at a small angle to the longer diagonal of the end-face. One half is turned through 180^ about an axis at right angles to this plane, and the two are reunited. If a narrow beam of parallel rays fall normally on one end of such an arrangement, the ordinary rays travel straight through without deviation, but their planes of polarisation in the two halves are inclined to each other at a small angle. The extraordinary rays are thrown off to either side of the apparatus, and if the prism be long enough and the beam not too wide, they can be separated entirely from the ordinary rays and stopped by a diaphragm with a small circular hole in it through which the ordinary rays pass. (2) Cornu's prism : — A Nicol or other polarising prism is taken and cut in two by a plane parallel to its length. A wedge-shaped piece is cut off one half, the edge of the wedge being parallel to the length of the prism, and the angle of the wedge some 3**. The two are then reunited, thus forming two half-Nicols, with their principal planes inclined at a small angle. The light emerging from each half is plane- polarised, the planes being inclined at a small angle. Both of these suffer from the defects that the angle between the planes of polarisation is fixed and that the sur- face of separation of the two halves being considerable, unless the incident light is very strictly parallel, some is reflected from this surface, and hence the line of separation is indistinct and ill-defined. (3) Lippich*s arrangement : — The polariser is a Glan's prisnL Lippich finds this more Ch, XV. S 65. J Polarised Light. 415 convenient than a Nicol, because of the lateral displacement of the light produced by the latter. A second Clan's prism is cut in two by a plane parallel to its length, and placed so that half the light from the first prism passes through it, while the other half passes at one side. The first prism is capable of rotation about an axis parallel to its length, and is placed so that its principal plane is inclined at a small angle, which can be varied at will, to that of the half-prism. The plane of polarisation of the rays which emerge from this half-prism is therefore slightly inclined to that of the rays which pass to one side of it, and this small angle can be adjusted as may be requited This arrangement also has the disadvantage that the surface of separation is large, and therefore the line of division is apt to become indistinct. (4) Lippich has used another arrangement, which re- quires a divided lens for either the telescope or collimator, and is, in consequence, somewhat complicated, though in his hands it has given most admirable results. All these four arrangements can be used with white light, and are therefore convenient in all cases in which the rotatory dispersion produced by the active substance, due to variation of wave-length in the light used, is too small to be taken into account (5) Laurent's apparatus : — The polariser is a Nicol followed by a half-wave plate for sodium light, made of quartz or some other crystal If quartz cut parallel to the axis be used, the thickness of the plate will be an odd multiple of '0032 cm. One of the axes of this plate is inclined at a small angle to the principal plane of the Nicol. The plate is semicircular in form and covers half the field — ^half the light passes through it, the other half to one side. The light on emerging from the plate is plane-polarised, and its plane of polarisation is in* dined to the axis of the quartz at the same angle as that of 41 6 Practical Physics. [Ch. XV. §#5 the incident light, but on the opposite side of that axis. Wc have thus plane-polarised light in the two halves of the field — the angle between the two planes of pdarisadon being small And, again, by varying the angle between the axis of the quartz and the plane of polarisation of the incident light, we can make the angle between the planes of polarisa- tion in the two halves of the field anything we please ; but, on the other hand, since the method requires a half-wave plate, light of definite refrangibility must be used. (6) Poynting's method : — Poynting suggested that the desired result might be obtained by allowing the light from one half the field, afler traversing a NicoFs prism, to pass through such a thickness of some rotatory medium as would suffice to produce in its plane of polarisation a rotation of 2** or 3**. If quartz cut perpendicular to the axis be used, this will be about •01 cm. for sodium light A plate of quartz so thin as this being somewhat difficult to work, Poynting suggested the use of a thicker plate which had been cut in two ; one half of this thicker plate is reduced in thickness by about •01 cm., and the two pieces put together again as before ; the light from one half the field traverses 'oi cm. of quartz more than the other, and hence the required effect is pro- duced. This works well, but it is important that the light should pass through both plates of quartz parallel to the axis, otherwise elliptic polarisation is produced. Moreover, the difficulty of obtaining a plate of quartz *oi cm. thick is not really very great Another suggestion of Poynting*s was to use a glass cell with a solution of sugar or other active substance in it A piece of plate glass of 3 or 4 mm. in thickness is placed in the cell, the edge of the plate being flat and smooth. The polarised light from half the field passes through the glass plate, that from the other half traverses an extra thickness of some 3 or 4 mm. of sugar solution, which rotates it through Cu. XV. § 65.1 Polarised Ligki, 417 the required angle. This method has an advantage over the quartz that we are able to adjust the angle between the l>lanes of polarisation in the two halves of the field by vary- ing the strength of the solution. Its simplicity is a strong point in its favour. It has the disadvantage that it is rather difficult to get a dear sharp edge, but care overcomes this. Of course the adjustments necessary in the position of the Nicols, the method of taking the readings, &c., are the same as those in the last section. Experiment, — Set up a half-shadow polarimeter and mea- sure the rotation produced in active solutions of various strengths, determining the relation between the strength of the solution and the rotation. Enter results as in preceding section. CHAPTER XVI. COLOUR VISION,' 66. The Colour Top. This apparatus consists of a spindle, which can be rapidly rotated by means of a pulley fixed to it, and from this a string or band passes to the driving wheel of some motor.* A disc whose edge is graduated in one hundred parts turns with the spindle, and by means of a nut and washer on the end of the spindle, coloured discs can be fixed against this divided circle. From six coloured papers — black, white, red, green, yellow, and blue — discs of two sizes are prepared and are then slit along a radius from circum- ference to centre so as to admit of being slipped one over the other. Each has a hole at the centre through which the spindle can pass. The apparatus is arranged to shew that, if any five out 1 See Deschanel, Natural Philosophy y chap. Ixiii. * The water motor referred to in § 28 is very convenient for thlfl Qieperiment« K £ 41 8 Practical Physics. [Ch. xvi. §«. of these six discs be taken, a match or colour equation between them is possible. For instance, if yellow be ex- cluded, the other five may be arranged so that a mixture of red, green, and blue is matched against one of black and white. Take, then, the three large discs of these colours and, slipping them one on the other, fix them against the graduated circle. Start the motor and let it rotate rapidly, looking at the discs against a uniform background of some neutral tint The three colours will then appear blended into one. Now place the small discs on these; then on rotating the whole, it will be found that the white and black blend into a grey tint By continual adjustments an arrangement may be found, after repeated trials, such that the colour of the inner circle is exactly the same both in tint and luminosity as that of the outer ring. The quantities of colour exposed may then be read off on the graduated circle, and it will be found that the proportions are some- what like the following : 79 parts blade and 21 white match 29*2 blue, 29*2 green, and 41*6 red. With the six discs six equations of this kind can be formed leaving out each colour in tiuiL But, according to Maxwell's theory of colour, a match can be found between any four colours, either combining them two and two in proper proportions, or one against three. The colour top is not suited to shew this, for with it we have another condition to fulfil. The whole circumference of the circles has in each case to be filled up with the discs. The vacant spaces must therefore be filled up with black, which alters the intensity of the resultant tints; but the intensity niay be adjusted by altering the sizes of all the coloured sectors proportionately, and hence with any four colours and black a match can be made. And thus from the theory the six final equations are not independent; for between any four of the variables, the colours, there exists a fixed definite relation. I^ then, we take two of the equations, we can by a simple algebraical calculation find the others, A comparison between tbc Ch. XVI. §66.] Colour Vision. 419 equations thus formed and those given directly as the re- sult of the experiments forms a test of the theory; but in practice it is better, in order to insure greater accuracy, to combine all the equations into two, which may then be made the basis of calculation, and from which we may form a second set of six equations necessarily consistent among themselves and agreeing as nearly as is possible with the observations. A comparison between these two sets gives evidence as to the truth of the theory, or, if we consider this beyond doubt, tests the accuracy of the observations. The six equations referred to are formed from the six found experimentally by the method of least squares. Thus let us denote the colours by the symbols x^ y^ z^ u, v, Wy and the quantities of each used by a„ ^|, ^i, ^1, ^1, fx in the first equation, and by the same letters with 3, 3, &c, subscript in the others, and let 2 {x} denote the sum formed by adding together a series of quantities such as x. Our six equations are And w« have to make ^{ax-{-by-{'Cs-{'du-{'ev-{'Jw)^ a minimum, treating x, y^ z, u, v, w as variables. The resulting equations will be the following : — xXa^-^-yiad-^-z^ac+uliad +vlta€'{-wXa/= o, xl,da+yi^+zidc-^uidd +v'S^€-¥wid/= o. &C. &C. The calculation of the six equations in this manner is a somewhat long and troublesome process, while the numbers actually arrived at will depend greatly on the exact colours of the discs. In a paper on the subject (* Nature,' Jan. 19, 187 1), from which the above account is taken. Lord Rayleigh calls attention to the importance of having the discs accurately cut and centred, otherwise on rotation a E B 2 420 Practical Physics. [Ch. XVI. § 66 coloured ring appears between the two uniform tints and gives rise to difficulty. The results also depend to a very considerable ext^t upon the kind of light with which the discs are illuminated The difference between light from a cloudless blue sky and light from the c)ouds is distinctly shewn in the numbers recorded in the paper referred to above. The numbers obtained may also be different for different observers; the experiment, indeed, forms a test of the colour- perception of the observer. At the Cavendish Laboratory the colour top is driven by a small water turbme by Baily & Co., of Manchester. The following table is taken from Lord Rayleigh's paper, being the record of his experiments on July 20, 1870. The circle actually used by him had 192 divisions ; his numbers have been reduced to a circle with 100 divisions by multi- plying them by 100 and dividing by 193. The second line in each set gives the results of the calculations, while m the first the observed numbers are recorded. Table. Black WhUe Red Green Yellow Blue + 15-6 + I6-I + 608 + 60-4 + 236 + 235 -411 -41-5 -58-9 -58-5 + 468 + 447 -66-6 -66-8 -33'4 -33-2 + 29-1 + 29*6 + 24-1 + 257 -707 -71-2 -29-3 -28-8 + 11*4 + II-6 + 27 + 27 + 61-6 + 61*4 + 51*6 + 26 + 265 + 22 + 21*9 -333 -33-8 -667 -66-2 -79 -79*3 -21 -207 + 41-6 + 42-1 + 29*2 + 29*2 -I-29-2 + 287 + 70-2 + 706 + 10-9 + 11*3 -64 -638 -36 -36-2 + 189 + 181 Ch. XVL § 66,J Colour Vision. 421 Experiment~^Y orm a series of colour matches with the six given discs, taken five at a time, and compare your results with those given by calculation. Enter the results as in the above table. 67. The Spectro-photometer. This instrument consists of a long, flat rectangular box iSi%. 41). At one end of this there is a slit, a, the width of which can be adjusted. The white light from a source Fig. 4z. behind the slit passes through a colliroating lens, l, placed at the distance of its own focal length from a, and falls as a parallel pencil on the set of direct- vision prisms ss'. The emergent beam is brought to a focus by the second lens m, and a pure spectrum thus formed at the end of the box. A sliding-piece fitted to this end carries a narrow slit b, through which any desired part of the spectrum may be viewed, c is a second slit, illuminated also by white light, the rays from which after passing through the lens n fall on a plane mirror k, and being there reflected traverse the prisms and form a second spectrum directly below the flrst By adjusting the positions of the lenses and the mirror k the lines in the two spectra can be made to coincide. The light from a passes over the top of the mirror and the two spectra are seen one above the other. A concave lens enables the observer to focus distinctly the line of sep)aration at k. In front of the three slits respectively are three Nicol's 422 Practical Physics. [Ch. XVI. $ 67. prisms, F, G, H. F is fixed with its principal plane vertical, parallel, therefore, to the slits and edges of the prisms ; g has its principal plane horizontal, while h is capable of rotation round a horizontal axis parallel to the length of the box ; p is a pointer fixed to the prism h and moving over a graduated circle q r, which is divided into 360 parts. The zero of the graduations is at the top of the circle, and when the pointer reads zero the principal plane of h is vertical The Nicols f and g polarise the light coming through the slits, the first in the horizontal plane, the second in the vertical The emergent beam is analysed by the Nicol h. When the pointer reads zero or 180** all the light in the upper spectrum from the slit a passes through h, but none of that from c is transmitted. As the Nicol is rotated through 90^ the quantity of light from a which is transmitted decreases, while the amount coming from c increases, and when the Nicol has been tinned through 90^ all the light from c is transmitted and none from a. For some position then between o and 90** the bright- ness of the small portions of the two spectra viewed will be the same. Let the reading of the pointer when this is the case be 0. Let the amplitude of the disturbance from a be Uy that of the disturbance from c be ^, then clearly a cos ^ = ^ sin ^, and if I« I, be the respective luminous intensities, J^=^'^ tan« 0, Now place anywhere between l and k a small rectan- gular cell containing an absorbing solution. The upper spectrum will become darker and the Nicol will require to be moved to establish equality again in the brightness. Let ff be the new reading, and V^ the intensity of the light which now reaches the eye from a. Then * ~-= tan' &, ' See Glaiebrook, Physical Optics^ pp. ia-27. Ch. XVL §67.1 Colour Vision. 423 Thus But if k represent the fraction of the light lost by absorp- tion and reflexion at the faces of the vessel, we have r,=i.(i->j). Hence . tan*^ tan^tf To eliminate the effects of the vessel the experiment should be repeated with the vessel filled with water or some other fluid for which the absorption is small ; the difference between the two results will give the absorption due to the thickness used of the absorbing medium. Of course in all cases four positions of the Nicol can be found in which the two spectra will appear to have the same intensity. At least two of these positions — ^which are not at opposite ends of the same diameter — should be observed and the mean taken. In this manner the index error of the pointer or circle will be eliminated. For success in the experiments it is necessary that the sources of light should be steady throughout In the experi- ments recorded below two argand gas-burners with ground- glass globes were used The apparatus and burners must remain fixed, relatively to each other, during the observa- tions.' Dr. Lea has recently suggested another method of using the instrument to compare the concentration of solutions of the same substance of different strengths. A cell is employed with parallel faces, the distance between which can be varied at pleasure. A standard solu- tion of known strength is taken and placed in a cell of known thickness ; let c^ be the concentration, that is, the > See Proc, Cam, Phil, Soe,^ toL it. Part VI. (Glazebrook on ^ Spectro-photometer). 424 Practical Physics, [Ch. XVI. § 67. quantity of absorbing matter in a unit of volume, m^ the thickness of this solution. The apparatus is adjusted until the intensity in the two images examined is the same. The other solution of the same medium is put in the adjustable cell, which is then placed in the instrument, the standard being removed, and the thickness is adjusted, without altering the Nicols, until the two images are again of the same intensity, whence, if ^ be the concentration, m the thickness, we can shew that cm = C\ fit\ \ /. c=Cimilm ...... (i) and from this c can be found, for all the other quantities are known. We may arrive at equation (i) from the following simple considerations. If ^ be the concentration, cm will be propor- tional to the quantity of absorbing material through which the light passes. I( then, we suppose that with the same absorbent the loss of light depends only on the quantity of absorbing matter through which the light passes, since in the two cases the loss of light is the same, we must have cm = ^1^1, or c = CiMilm, Experiments, (i) Determine by observations in the red, green, and blue parts of the spectrum the proportion of light lost by passing through the given solution. (2) Determine by observations in the red, green, and blue the ratio of the concentration of the two solutions. Enter results thus : — (i.) Solution of sulphate of copper i cm. in thickness. Colour % ¥ k Red, near C . , . Green, near F . - . Blue-green. 60^*50' 61° 30' 64^30' 49" 50' 56° 30' 58° 30' •56 •33 •39 Cm. XVI. J 67.] Colour Vision. 425 (3.) Two solutions of sulphate of copper examined. Stan- dara solution, 10 per cent, i cm. in thickness. Thickness of experimental solution giving the same absorp- tion observed, each mean of five observations. Colour of Light Thickness Ratio of Concentrations Blue .... Green . . • . Red .... 74 73 75 1*35 1-37 1-33 68. The Colour Box. The colour box is an arrangement for mixing in known proportions the colours from different parts of the spectrum and comparing the compound colour thus produced with some standard colour or with a mixture of colours from some other parts of the spectrum. Maxwell's colour box is the most complete form of the apparatus, but it is somewhat too complicated for an elemen- tary course of experiments. We proceed to describe a modification ol it, devised by Lord Rayleigh, to mix two spectrum colours together and compare them with a third. This colour box is essentially the spectro-photometer, described in the last section, with the two Nicols f and g removed Between the lens l and the mirror k is placed a double-image prism of small angle, rendered nearly achromatic for the ordinary rays by means of a glass prism cemented to it This prism, as well as the mirror k, is capable of adjustment about an axis normal to the bottom of the box. The prism thus forms two images of the slit, the apparent distance between which depends on the angle at whidi the light falls on the prism ; this distance can therefore be varied by turning the prism round its axis. The light coming from these two images falls on the direct-vision spectroscope ss', and two spectra are thus formed in the focal plane qr. These two spectra overlap, so that at any point, such as b, we have two colours mixed, one from each spectrum. The amount of overlapping 426 Practical Physics, [Ch. XVI. § 68. and therefore the particular colours which are mixed at each pointy depend on the position of the double-image prism, and, by adjusting this, can be varied within certain limit& Moreover, on passing through the double image prism the light from each slit is polarised, and the planes of polarisation in the two beams are at right angles. We will suppose that the one is horizontal, the other vertical Thus, in the two overlapping spectra the light in one spectrum is polarised horizontally, in the other vertically. For one position of the analysing prism the whole of one spectrum is quenched, for another position at right angles to this the whole of the second spectrum is quenched. The proportion of light, then, which reaches the eye when the two spectra are viewed, depends on the position of the analyser, and can be varied by turning this round. Thus, by rotating the analjTser we can obtain the colour formed by the mixture of two spectrum colours in any desired proportions, and at the same time the proportions can be calculated by noting the position of the pointer attached to the analyser. For if we call A and b the two colours, and suppose that when the pointer reads o'' the whole of the light from a and none of that from b passes through, and when it reads 90"^ all the Ught from b and none from a is transmitted, while a, ^ denote the maximum brightnesses of the two as they would reach the eye if the Nicol h were removed, then when the pointer reads ^ we shall have Intensity_ofB^a^^,^ Intensity of A ^ The standard light will be that in the lower part of the field, which comes from the slit c, after reflexion at the mirror k. This light being almost unpolarised — the re- flexions and refractions it undergoes slightly polarise it — is only slightly affected in intensity by the motion of the analyser. By adjusting the tap of the gas-burner we can alter its intensity, and by turning the mirror k we can bring any desired portion of the spectrum to the point a Cii. XVL] Colour Vision 427 rhe instrument was designed to shew that a pure yellow, «uch as that near the d line, could be matched by a mixture of red and green in proper proportions, and to measure those proportions. It is arranged, therefore, in such a way that the red of one spectrum and the green of the other overlap in the upper half of the field at b, while the yellow of the light from c is visible at the same time in the lower half. Experiment. — Determine the proportions of red and green li^ht required to match the given yellow. Enter results thus :— Values of d • « • • 59** (xP 15' 59" 45^ Mean • , . • . 60^ Ratio of intensities ^. /3 R. Colour Photometry. Captain Abney has recently shewn how, by a modifica- tion of Rumford's photometer, the luminous intensity at each point of the spectrum may be compared with that from a given source. For this purpose a pure spectrum of the given source is produced on a screen. This may be done as in chap, xiv., fig- 34- It is preferable, however, to use two lenses in such a way that the light from the slit Si (fig. xxxiii), which is placed at the. principal focus of the first lens, falls as a parallel beam on the prism p. After refraction through it, parallel rays of each different colour fall on the lens i^, and are brought by it to a focus on the screen d d. In this screen there is a second slit (Ss)) through which rays of only one refrangibility pass. These rays fall on a third lens (l,) arranged so as to produce on a white screen at f e an image of the nearer face of the 428 Practical Physics, [Ch. XVI. § R. prism. This image is illuminated only by light which has passed through 83 — that is, by light of a definite colour, and by moving the slit S2 a patch of light of any required colour can be thrown on to the screen at f e. The lenses used will not, in general, be achromatic, and thus the images of Si formed by the different colours will not be at the same distance from L2, but by tilting the screen dd they can all be brought into focus. Again, since the face of the prism P2 is not at right angles to the direction in which the light travels from It to reach the slit 82, the lens L3 is also slightly tilted in order to form on f e a sharp image of the whole of this face. To apply this to colour photometry, a vertical stick is placed in the path of this coloured beam, casting a shadow on the screen, while a second (standard) light (Tj), mounted on a scale, casts a second shadow close by. This second shadow is coloured, being illuminated by the coloured beam from 82, while the first shadow receives the light from the standard ; still, by moving the comparison light along the scale a point can be found at which the luminosities over the two appear equal. The determination of this point is, <=^' XVI. § R.] Colour Vision. 429 however, attended with some difficulty, much of which is overcome by the adoption of the following oscillation method, the account of which is taken from the Bakerian Lecture for 1886 by Sir William Abney and Major-General Festing. The illuminating value of the spectrum varies greatly in rts different parts, the maximum usually being in the yellow, and there is a gradation from this towards either end. Now suppose that widi the standard h'ght at, say, 50 cm. from the screen it is approximately of the same intensity as the yellow hght of the spectrum, then if the standard be moved to, say, 60 cm. distance there will be two parts of the spectrum, one towards the red the other towards the blue, which will have the same luminosity as the standard at a distance of 60 cm. ; this is, of course, ^5/3^ of its value when at 50 cm. To find these points, the card to which the slit s^ i& attached is movable, and the slit can be made to slide along the spectrum, its position being determined by means of a scale. When the standard is at 60 cm. distance and the slit in the yellow, the shadow of the rod illuminated by the white light will be palpably darker than the other ; when the slit has passed into the green-blue, it will be palpably lighter. SirWm. Abney finds * that the best way of determining the inter- mediate point where the shadows balance is by oscillating the slide gently between two points where first one shadow and then the other is palpably too dark; the oscillations become shorter and shorter until the point of balance \s determined.' The slide is dien moved through the yellow to the red, and the same process \s repeated. Two points in the spectrum whose illumination corre- sponds to that of the standard at the distance of 60 cm. are thus found. This distance is then varied, and another pair of points determined. In this manner a curve \s drawn in which the abscissa represent the position of the slit, while 430 Practical Physics, [Ch. XVI. § R. the ordinates give the intensity of the light in tenns of that of the standard. By means of an independent series of observations the wave-length of the light which falls on the slit in any given position can be found, as in § 62, and thus a curve giving intensity in terms of wave-length can be determined. This curve is called a luminosity curve. The form of the curve, as found by Sir Wm. Abney, is given in fig. xxxiv. The ^L4000 C F 5000 E D 6000 C B 7000 measurements are to some extent affected by the colour of the receiving screen ; a card painted with two coats of zinc oxide gives the best results. A portion of this screen about 5 cm. square, limited by a sheet of black paper with a hole cut in it, should be used. Instead of moving the standard light, the method of varying its intensity adopted by Sir Wm. Abney in some later experiments may be employed (* Proc. R. S.' vol xliii. p. 249). A circular disc is placed between the standard light and the screen. The disc is divided into four quadrantal sectors, and the alternate sectors are removed. If sucb a disc is rotated between the light and the screen, it is clear that half the light is cut off. To the disc a pair of movable sectors are fitted, and these can be adjusted so as to close Ch. XVI. § R.] Colour Vision. 431 to a greater or less extent, as may be required, the open sectors of the main disc. If, for example, the open sectors be half closed by the adjustable sectors, the transmitted light has only half the intensity of that previously trans- mitted. By means of suitable mechanism the position of these movable sectors can be adjusted relatively to the others while the apparatus is in motion, and thus the amount of light from the standard can be varied until the luminosity of the shadowy is the same. In this method of making the observations the slit is fixed in position and the sectors adjusted. When the adjustment has been made the motor is stopped, and the position of the sectors determined ; from this the intensity of the standard can be found. The apparatus can be used to examine the effect of colour mixtures by placing two or more slits in the screen d d. A coloured image of the face of the prism will be formed by light passing through each slit, and these images are super- posed. By opening each slit in turn and finding the luminosity, and then making measurements with the two or three slits open simultaneously, we can verify the law that the impression due to a mixed light is the sum of the im- pressions due to each light separately. The apparatus has been employed by Sir Wm. Abney to study colour-blindness, by comparing the luminosity curves found by various observers, and also for experiments on the scattering of light by small particles. For this purpose a glass trough filled with pure water was placed between the source and the slit Si, and the luminosity curve found. Then a solution of mastic in alcohol was dropped in various quantities into the water, and the curve again determined. It was found that the intensity of the transmitted light was very closely in accordance with the formula found by Lord Rayleigh, in accordance with which I = Io^ -ijpV 432 Practical Physics, [Ch. XVI. § R. lo being the intensity of the incident light, x the thick- ness of the absorbing medium, kz. constant, and X the wave- length. Experiments. (i) Determine the luminosity curve for the various com- ponents of the light from the given source, and compare the result with the normal curve. (2) Shew that the intensity of a mixture of colours is the sum of the intensities of the components. (3) Determine the absorption in different parts of the spectrum produced by (he given solution of mastic, and com- pare your result with Lord Rayleigh's formula. Enter results thus : — (I) Scale Reading Inteimty Scale Reading Intensity 60 •35 48^ 80 564 1*2 47-8 97 53-6 4*2 47-1 100 52 9-6 46-9 100 51 19*4 46-2 96 50 43*5 45*4 82 49*2 73 44-9 59 44 16 42-4 The curve can be drawn from these. (2) Slits were placed in the red, green, and violet, and the Slit Open Observed Calcolated R 203 204-25 (R+G) 242 24175 G 38-5 37'5 (c + v) 45-0 46*0 V 8-5 8-5 (k + V) 214*0 212-5 (R + G + V) 250X) 250*25 luminosities observed for each slit separately, and for the slits ClL XVI. § R.] Colour Vision, 433 ill pairs, and also all three together. The corresponding values were calculated from the curve on the assumption that the resulting impression is the sum of the individual ones. (3) The intensity for various wave-lengths before and after absorption was determined. The table gives the observed and calculated ratio : — Wave Length Observed Ratio Calculated Ratio 6448 13*1 127 6374 12*1 12-3 6210 11-85 1 1-6 5900 10 9.9 5589 8-25 81 5459 7*4 7'3 5180 5-6 5-6 4602 4-8 48 CHAPTER XVII. MAGNETISM. Properties of Magnets, Certain bodies, as, for instance, the iron ore called lode- stone, and pieces of steel which have been subjected to certain treatment, are found to possess the following pro- perties, among others, and are called magnets. If a magnet be suspended at any part of the earth's sur- face, except certain so-called magnetic poles, so as to be free to turn about a vertical axis, it will in general tend to set itself in a certain azimuth — />. with any given vertical plane, fixed in the body, inclined at a certain definite angle to the geo- graphical meridian — and if disturbed from this position will oscillate about it If a piece of iron or steel, or another magnet, be brought FF 434 Practical Physics, [Ch. XVLL near to a magnet so suspended, the latter will be deflected from its position of equilibrium. If a magnet be brought near to a piece of soft irou or unmagnetised steel, the iron or steel will be attracted by the magnet. This is illustrated by the experiment of § S, p. 467. If a long thin magnetised bar of steel be suspended so as to be capable of turning about a vertical axis through its centre of gravity, it will be found to point nearly north and south. We shall call the end which paints north the north end of the magnet^ the other the south end. Definition of Uniform Magnetisation. — If a magnet be broken up into any number of pieces, each of these is found to be a magnet Let us suppose that the magnet can be divided into a very large number of very small, equal, similar, and similarly situated parts, and that each of the parts is found to have exactly the same magnetic properties. The magnet is then said to be uniformly magnetised. Definition of Magnetic Axis of a Magnet. — If any magnet be supported so as to be free to turn in any direc- tion about its centre of gravity, it is found that there is a certain straight line in the magnet which always takes up a certain definite direction with reference to the earth. This line is called the magnetic axis of the magnet. Definition of Magnetic Meridian. — ^The vertical plane through this fixed direction is called the plane of the magnetic meridian. Definition of Magnetic Poles. — If the magnet be a long thin cylindrical bar, uniformly magnetised in such a way that the magnetic axis is parallel to the length of the bar, the points in which the axis cuts the ends of the bar are the magnetic poles. The end of the bar which tends to point north, when the magnet is freely suspended, is the norths or positive pole \ the other is the souths or negative pole. Such a magnet is called solenoidal, and behaves to other magnets as if the poles were centres offeree, the rest of the magnet being devoid of magnetic action. In all actual Ch. XVII.] Magnetism. 435 magnets the magnetisation differs from uniformity. No two single points can strictly be taken as centres of force com- pletely representing the action of the magnet. For many practical purposes, however, a well-made bar magnet may be treated as solenoidal with sufficient accuracy. A Robison mag- net consisting of two small steel spheres connected by a piece of steel wire will be found convenient for many experiments. The poles in this case are practically at the centres of the spheres. The following are the laws of force between two mag- netic poles : — (i) There is a repulsive force between any two like mag- netic poleSy and an attractive force between any two unlike poles, (2) The magnitude of the force is in each case numerically equal to the product of the strength of the poles divided by the square of the distance between them. This second law is virtually a definition of the strength of a magnetic pole. In any magnet the strength of the positive pole is equal in magnitude, opposite in sign, to that of the negative pole. If the strength of the positive pole be w, that of the negative pole is — m. Instead of the term * strength of pole,' the term • quantity of magnetism ' is sometimes used. We may say, therefore, that the uniformly and longitudinally magnetised thin cylindrical bar behaves as if it had quantities m and —m of magnetism at its two ends, north and south respectively ; we must, however, attach no properties to magnetism but those observed in the poles of magnets. If, then, we have two magnetic pdles of strengths m and m\ or two quantities of magnetism m and m\ at a distance of r centimetres apart, there is a force of repulsion between them which, if m and m' are measured in terms of a proper unit, is mm* fr'^ dynes. If one of the two m or m* be negative, the repulsion becomes an attraction. The C.G.S. unit strength of pole is that of a pole which F F 2 436 Practical Physics. [Ch. xvri. repels an equal pole placed a centimetre away in air with a force of one dyne. In practice it is impossible to obtain a single isolated pole ; the total quantity of magnetism in any actual magnet, reckoned algebraically, is always zero. Definition of Magnetic Field. — ^A portion of space throughout which magnetic effects are exerted by any distri- bution of magnetism is called the magnetic field due to that distribution. At each point of a magnetic field a pole of strength m is acted on by a definite force. The Resultant Magnetic Fora at each point of the field is the force which is exerted at that point on a positive pole of unit strength placed thera This is also called the Strength of the Magnetic Field at the point in question. The strength of a magnetic field is measured in ' Gausses ' ; a field of one ' Gauss ' has unit strength. If H be the strength of the field, or the resultant mag- netic force at any point, the force actually exerted at that point on a pole of strength m\&mYL. The magnetic force at each point of the field will be definite in direction as well as in magnitude. Definition of Line of Magnetic Force. — If at any point of the field a straight line be drawn in the direction of the magnetic force at that point, that straight line vdll be a tangent to the Line of Magnetic Fora which passes through the point A Line of Magnetic Force is a line drawn in such a manner that the tangent to it at each point of its length is in the direction of the resultant magnetic force at that point A north magnetic pole placed at any point of a line of force would be urged by the magnetic force in the direc- tion of the line of force. As we shall see shortly, a small magnet, free to turn about its centre of gravity, will place itself so that its axis is in the direction of a line of force. A surface which at each point is at right angles to the Ch. XVil.] Magnetism. 437 line of force passing through that point is called a level surface or surface of equilibrium, for since the lines of force are normal to the surface, a north magnetic pole placed anywhere on the surfoce will be urged by the magnetic forces perpendicularly to the surface, either inwards or outwards, and might therefore be regarded as kept in equi librium by the magnetic forces and the pressure of the surface. Moreover, if the pole be made to move in any way over the surface, since at each point of its path the direction of its displacement is at right angles to the direction of the resultant force, no work is done during the motion. Definition of Magnetic Potential. — ^The magnetic potential at any point is the work done against the magnetic forces in bringing up a unit magnetic pole from the boundary of the magnetic field to the point in question. The work done in transferring a unit magnetic pole from one point to another against magnetic forces is the difference between the valties of the magnetic potential at t/iose points. Hence it follows that the magnetic potential is the same at all points of a level surface. It is therefore called an equi- potential surface. Let us suppose that we can draw an equipotential surface belonging to a certain configuration of magnets, and that we know the strength of the magnetic field at each point of the surface. Take a small element of area, a square centimetres in extent, round any point, and through it draw lines of force in such a manner that if h be the strength of the magnetic field at the point, the number of lines of force which pass through the area a is h a. Draw these lines so that they are uniformly distributed over this small area. Do this for all points of the surface. Take any other point of the field which is not on this equipotential surface ; draw a small element of a second equipotential surface round the second point and let its area be a' square centimetres. This area will, of course, be per- 43^ Practical Physics. (Ch. XVIL pendlcuhr to the lines of force which pass through it Suppose that the number of lines of force which pass through this area is n\ then it can be proved, as a conse- quence of the law of force between two quantities of mag- netism, that the strength of the field at any point of this second small area a! is numerically equal to the ratio nffa'. The field of force can thus be mapped out by means ot the lines of force, and the intensity of the field at each point determined by their aid. The intensity is numerically equal to the number of lines of force passing through any small area of an equipotential surface divided by the number of square centimetres in that area, provided that the lines of force have originally been drawn in the manner described above.' > For an explanation of the method of mapping a field of force by means^ of lines of force, see Maxwell's Elementary Electricity y chaps, v. and vi. and J. J. Thomson's Elements of Electricity^ and Magnetism^ chap. ii. The necessary propositions may be summarised thus (leaving out the proofs^ : — (i) Consider any closed surface in the field of force, and imagine the sur&ce divided up into very small elements, the area of one of which is 0* ; let F be the resultant force at any point of «*, resolved nonnally to the surface inwards ; let 2 F 0* denote the result of adding together the products F 9 for every small elementary area of the dosed surface. Then, if the field of force be due to matter, real or imaginary, for which the law of attraction or repulsion is that of the inverse square of the distance, 2F0-«4irM, where M is the quantity of the real or imaginary matter in question contained inside the closed sur&ce. (2) Apply proposition (i) to the case of the closed surface formed by the section of a tube of force cut off between two equipotential sur* foces. [A tube of force is the tube formed by drawing hnes of force through every point of a closed curve.] Suppose 9 and & are the areas of the two ends of the tube, F and f' the forces there ; then F 0* « F V. (3) Imagine an equipotential surface divided into a large number of very smaU areas, in such a mimner that the force at any point is in- versely proportional to the area in which the point falls. Then 9 being the measure of an area and F the force there, F 0* is constant for every clement of the surface. (4) Imagine the field of fbrce filled with tubes of force, with the elementary areas of the equipotential sur&ce of proposition (3) as bases. "hese tubes will cut a second equipotential surface in a series of de- ntary areas ff'. Let f' be force at 0', then by propositions (2) and Ch. xvii.j Magnetism. 439 On the magnetic potential due to a single pole. — ^The fcKce between two magnetic poles of strengths m and ///', at a distance r^ centimetres apart is, we have seen, a re- pulsion ofmrn^/ri^dynes. Let us su^^ose the pole /»' moved towards m through a small distance. Let a (fig. 42) be ^ p, p^ p the position of f«, Pi, Pjthe ' ' ' ' two positions of m\ Then a P3 Pj is a straight line, and A P| =ri. Let AP9 = r2, PiPa^^^^a* Then, if, during the motion, from Pj to P2y the force remained constant and of the same value as at Pi, the work done would be ^^ (-.-,); while if, during the motion, the force had retained the value which it has at P2, the work would have been mm' r(^i~^a)- Thus the work actually done lies between these two values. But since these fractions are both very small, we may neglect the difference between ri and rj in the denominators. Thus the denominator of each may be (3) f' ir' is constant for every small area of the second equipotential surface, and equal to F o-, and hence F 0- is constant for every section of every one 01 the tubes of force ; thus F <r= k. (5) By properly choosing the scale of the drawing, k may be made equal to unity. Hence F=a-, or the force at any point is equal to the number of tubes of force passing through the unit of- area of the equipotential surface which contains the point. (6) Each tube of force may be indicated by. the line of force which forms, so to speak, its axis. With this extended meaning of the term * line of force ' the proposition in the text follows. The student will notice that, in the chapter referred to, Maxwell very elegantly avoids the analysis here indicated by accepting the method of mapping the electrical field as experimentally verified, and deducing from it the law of the inverse square. 440 Practical Physics. [Ch. xvTi written r^r^ instead of r|* and r^ respectively. The two expressions become the same, ^d hence the work done is mm - * — % or mm' f — — 1\ \^2 rj Similarly the work done in going from p, to a third point, P3, is mm'(l^L\ \ri rj And hence we see, by adding the respective elements together, that the work done in going from a distance r' tP a distance r is "" G-.-) Hence the work done in bringing the pole m from infinity to a distance r from the pole m is mm' jr. But the potential due to f» at a distance r, being the work done in bringing up a unit pole from beyond the influence of the pole m^ will be found by dividing this by /»' ; it is therefore equal to mjr. Again, it follows from the principle of conservation of energy that the work done in moving a unit pole from any one point to any other is independent of the path, and hence the work done in moving the unit pole from any point whatever at a distance t* to any point at a distance r from the pole m is m a-j) For a single pole of strength m^ the equipotential sur- faces are clearly a series of concentric spheres, with m as centre ; the lines of force are radii of these spheres. Ch. XVILJ Magnetism. 441 If we have a sclenoidal magnet of strength m^ and ri, r^ be the distances of any point, p (fig. 43), from the positive and negative poles n and s no. 43. of the magnet, then the po- tential at p due to the north pole is «r/ri, and that due to the south pole is ^nt\r^\ hence the potential at p due to the magnet is F The equipotential surfaces are given by the equation > where ^ is a constant quantity, and the lines of force are at right angles to these surfaces. To find the resultant mag- netic force at p we have to compound a repulsion of fn\r^ along N p with an attraction of fn\r^ along p s, using the ordinary laws for the composition of forces. Let us now consider the case in which the lines of force in the space in question are a series of parallel straight lines uniformly distributed throughout the space. The intensity of the field will be the same throughout ; %uch a distribution constitutes a uniform magnetic field. The earth is magnetic, and the field of force which it pro- duces is practically uniform in the neighbourhood of any point provided that there be no large masses of iron near, and the lines of force are inclined to the horizon in these latitudes at an angle of about 67''. On the Forces on a Magnet in a Uniform Field. We proceed to investigate the forces on a solenoidal magnet in a uniform field. Let us suppose the magnet held with its axis at right angles to the lines of force, and let / be the distance between its poles, m the strength of each pole, and h the intensity 442 Practical Physics. [Ch. XVii. of the field. The north pole is acted on by a force mn 9X right angles to the axis of the magnet, the south pole by an equal, parallel, but opposite force m h. These two forces constitute a couple; the distance between the lines of action, or arm of the couple, is /, so that the moment of the couple is M / H. If the axis of the magnet be inclined at an angle to the lines of force, the arm of the couple will be m /sin ^, and its moment mlB. sin0. In all cases the couple will depend on the product ml. Definition of Magnetic Moment of a Magnbt. — The product of the strength of either pole into the distance between the poles, is called tlie magnetic moment of a solenoidal magnet. Let us denote it by m ; then we see that if the axis of the magnet be inclined at an angle to the lines of force, the couple tending to turn the magnet so that its axis shall be parallel to the lines of force is m h sin 6. Thus the couple only vanishes when 6 is zero ; that is, when the axis of the magnet is parallel to the lines of force. But, as we have said, the actual bar magnets which we shall use in the experiments described below are not strictly solenoidal, and we must therefore consider the behaviour, in a uniform field, of magnets only approximately solenoidd. If we were to divide a solenoidal magnet into an in- finitely large number of very small, equal, similar, and similarly situated portions, each of these would have identical magnetic properties ; each would be a small magnet with a north pole of strength m and a south pole of strength — /w. If we bring two of these elementary magnets together so as to begin to build up, as it were, the original magnet, the north pole of the one becomes adjacent to the south pole of the next; we have thus superposed, a north pole of strength m and a south pole of strength — /»; the e£fects of the two at any distant point being thus equal and opposite, no ex- ternal action can be observed. We have therefore a magnet equal in length to the sum of the lengths of the other two, with two poles of the same strength as those of either. Ch. XVII.] Magnetism. 443 14 however, wc were to divide up an actual magnet in this manner, the resulting elementary magnets would not all have the same properties. We may conceive of the magnet, then, as built up of a number of elementary magnets of equal volume but of dif- ferent strengths. Consider two consecutive elements, the north pole of the one of strength m is in contact with the south pole of the other of strength —»»' say ; we have at the point of junction a north pole of strength m-^m'^ we cannot replace the magnet by centres of repulsive and attractive force at its two ends respectively, and the calculation of its action becomes difficult I^ however, the magnet be a long bar of well-tempered steel carefully magnetised, it is found that there is very little magnetic action anywhere except near the ends. The elementary magnets of which we may suppose it to consist would have equal strengths until we get near the ends of the magnet, when they would be found to fall off somewhat The action of such a magnet may be fairly represented by that of two equal poles placed close to, but not coincident with, the ends ; and we might state, following the analogy of a solenoid, that the magnetic moment of such a magnet was measured by the product of the strength of either pole into the distance between its poles. We can, however, give another definition of this quantity which will apply with strictness to any magnet The moment of the couple on a solenoidal magnet, with its axis at an angle to ^e lines of magnetic force in a field of uniform intensity h, is, we have seen, m h sin ^, m being the mag- netic moment Thus the maximum couple which this magnet can experience is m h, and the maximum couple which the magnet can be subjected to in a field of uniform force of intensity unity is m. Now any magnet placed in a uniform field of magnetic force is acted on by a couple, and we may say that for 444 Practical Physics. [Ch. XVIL any magnet whatever, the magnetic moment of a nuzgnet is measured by the maximum couple to which the magnet can be subject when placed in a uniform magnetic field of intensity unity. The couple will be a maximum when the magnetic axis of the magnet is at right angles to the lines of force. If the angle between the axis of the magnet and the lines of force be ^, the magnetic moment m, and the strength of the field h, the couple will be m u sin 0, just as for a solenoidal magnet On the Potential due to a Solenoidal Magnet. We have seen that if p be a point at distances r^, r, from the north and south poles, n, s, respectively, of a solenoidal Fig. 44. _ magnet n o s (fig. 44) of strength m^ the magnetic potential at p is We will now put this ex- pression into another and iV^ more useful form, to which it is for our purposes practically equivalent. Let o, the middle point of the line n s, be the centre of the magnet ; let op = r, oN = os=/, so that 2/ is the length of the magnet, and let the angle between the magnetic axis and the radius vector o j> be 0, this angle being measured from the north pole to the south, so that in the figure N o p = A Draw N R, s T perpendicular to p o or p o produced, and suppose that op is so great compared with o n that we may neglect the square and higher powers of the ratio of o n/o p. Then p r n is a right angle, and p n R difiers very litde from a right angle, for o n is small compared with o p, so that p N = p R very approximately, and similarly p s = p T. Also OR~OT = ON cos P O N = I COS $. Ch. XVII.] Magnetism, 445 Thus r, = PN = po— OR = r— /cosfl = r [ I — - cos^Y and r<t = r+/cos ^ = r ( I + - cos ^Y and, if v denote the magnetic potential at p, we have ffl I I r I COS^ 1+ -COS^ [ r r ) tn r 2-COSfl r n „., : But we are to neglect terms involving /*/r*, etc. ; thus we may put If — ^ ^ ^r^r.^ ^ _ M COS ^ v = _5-cos^=— -^. if M be the moment of the magnet We shall see next how to obtain from this expression the magnetic force at p. On the Force due to a Solenoidal Magnet, To obtain this we must remember that the work done on a unit pole by the forces of any system in going from a point Pi to a second point p^, Vj, V2 being the potentials at P] and Pj, is v,— Vj. Let a be the distance between these two points, and let f be the average value of the magnetic force acting from Pj to Pq resolved along the line Pi Pj. Then the work done by the force Fin moving the pole is f tf. Hence Fa=V| — Vj, and if the distance a be sufficiently small, F,the average 44^ Practical Physics. [Ch, XVIL value of the foice between Pi and p^ may be taken as the force in the direction Pi p^ at either P] or Pj, Denoting it by f we have r=limiting value of ^^~^^ a when a is very small Let us suppose that Pi, P2 are two points on the same radius from o, that 0Pi=r and 0P2=r-h8. Fig. 45. Then is the same for th© ^^t two points, and we have „ MCOS^ V2= MCOS0 ('•+8)* MCOS g _MCOStf/ 28\ neglecting /^— j and higher powers (see p. 42). Also, in this case, a=& Thus F= limiting value of II — ^ a MCOS^/2S t^l /2d\__2MCOS^ We shall denote this by r, so that r is the force outwards, in the direction of the radius- vector, on a unit pole at a distance r from the centre of a small solenoidal magnet of moment m. If the radius-vector make an angle B with the axis of the magnet, we have 2M cos ^ lt=s Cn. XVII. 1 Magnetism. 447 Again, let us suppose that P1P2 (fig. 46) is a small arc of a circle with o as centre, so that ^ OP|=OP2=r ^^ **• let PjONsstf, and Pj0N=fl+<^. Thus The force, in this case, will be § that at right angles to the radius vector, tending to increase 0\ if we call it t we have T= —limiting value of ^^^l cr-^|cos(tf+t^)-costf| = -jSintf (see p. 45). These two expressions are approximately true ii the magnet n s be very small and solenoidal. We may dispense with the latter condition if the magnet be sufficiently small; for, as we have said, any carefully and regularly magnetised bar behaves approximately like a solenoid with its poles not quite coincident with its ends. In such a case 2 1 will be the distance between the poles, not the real length of the magnet, and 2 w / will still be the magnetic moment On the Effect on a Second Magnet In practice we require to find the efiect on two magnetic poles of equal but opposite strengths, not on a single ^^^' ^'* pole, for every magnet has two poles. Let us suppose that P (fig. 47) is the centre of A second magnet n'ps^ so small that we may, when considering the action of the distant magnet N o s, treat ft 448 Practical Physics. [Cn, XVII. as if either pole were coincident with p, that n^ is the strength, and 2 V the length of this magnet, and & the angle between p n' and o p produced. Then we have, acting outwards parallel to the radius vector o p on the pole n', a force 2 w' M cos B and an equal and parallel force acting inwards towards o on the pole s'; these two constitute a couple, the arm of which will be 2 /* sin &. Thus, if m' be the magnetic moment of the second magnet, so that m' = 2 »i' /', we have acting on this magnet a couple, tending to decrease ^, whose moment will be 2 MM^ co s B sin 9 This arises from the action of the radial force K. Tne tangential force on n' will be if tn' sin tf tending to decrease 0' and on s' an equal force also tending to decrease it These constitute another couple tending to decrease &\ the arm of this couple will be 2 /^ cos 0', and its moment will be M m' sin 6 cos 0' P Thus, combining the two, we shall have a couple, the moment of which, tending to increase O', will be -^- (sin cos e' + 2 sin 0' cos 0). It must of course be remembered that these expressions are only approximate ; we have neglected terms which, if the magnets are of considerable size, may become im- portant Ch. XVII.] Magnetism, 449 Two cases are of considerable interest and importance. In the first the axis of the first magnet passes through the centre of the second. The magnet n s is Fig.4«. said to be * end on.' 1 i — . >^ In this case (fig. 48) S o 14- ^ we have tf=o, and the action is a couple tending to decrease ^, the moment of which is 2 M m' ^. ^ — _- sm cr. If no other forces act on the second magnet, it will set itself with its axis in the prolongation of that of the first magnet In the second case (fig. 49) the line joining the centres of the two is at right angles to the axis of the first magnet, which is said to be ' broadside on '; then Fn. 49. = 90®, and we have a couple tending to ^1 increase ^, the moment of which will be \ P — =— cos U. "J We may notice that for a given value of r, the maximum value of the couple in this second case is only half of its maximum ._^ value in the former case. ft i The position of equilibrium will be that in which cos d' = o, or when the two axes are parallel Let us sup- pose that the second magnet is capable of rotating about a vertical axis through its centre, in the same way as a compass needle ; it will, if undisturbed, point north and south under the horizontal component of the magnetic force due to the earth ; let us call this h. Place the first magnet with its north pole pointing towards the second, and its centre exactly to the west of that of the second. The second will be defiected, \\s^ north pole turning to the east Let ^ be G G 450 Practical Physics. fCH. xvii. the angle through which it turns, then dearly ^ = 90 — ^ The moment of the earth's force on the magnet is m'h sin ^ that of the couple due to the other magnet is 2 M m' sin ^/r*, or 2 M m' cos ^/^, in the opposite direction. But the magnet is in equilibrium under these two couples, and hence we have m' h sm ^ = cos <^ Thus M = i^ H H tan 0. Next place the first magnet with its north pole west and its centre exactly to the south of the second ; the north pole of the second will move to the east through an angle ^, say, and in this case we shall have ^ = ^. The moment of the couple due to the earth wiU be as before m'h sin i/r ; that due to the first magnet is -3- cos^ and hence m = h r • tan ^. We shall see shortly how these formulae may be used to measure m and h. On the Measurement of Magnetic Force, The theoretical magnets we have been considering are all supposed to be, in strictness, simply solenoidal rods without thickness, mere mathematical lines in fact The formulae may be applied as a first approximation, however, to actual magnets, and we shall use them in the experiments to be described. There remains, Tiowever, for consideration the theory of an experiment wblch will enable us to compare the magnetic moments of a magnet of any form under different Ch. XVII.} Magnetisfn. 451 conditions of magnetisation, or of two magnets of known form, or to compare the strengths of two approximately uniform magnetic fields, or, finally, in conjunction with the formulse already obtained, to measure the moment of the magnet and the strength of the field in which it is. We have seen (p. 166) that, if a body, whose moment of inertia about a given axis is k, be capable of vibrating about that axis, and if the force which acts on the body after it has been turned through an angle 6^ from its position of equi- librium, tending to bring it back to that position, ht fxd^ then the body will oscillate isochronously about this posi- tion ; also if the time of a complete oscillation be t, then T is given by the formula = ="^Vu We shall apply this formula to the case of a magnet. We have seen already that, if a magnet be free to oscillate about a vertical axis through its centre of gravity, it will take up a position of equilibrium with its magnetic axis in the magnetic meridian. The force which keeps it in the meridian arises from the horizontal component of the earth's magnetic force ; and if the magnet be disturbed from this position through an angle ^, the moment of the couple tending to bring it back is m h sin Oy m being the magnetic moment Moreover, if be the circular measure of a small angle, we know that the difference between and sin d depends on ^ and may safely be neglected ; we may put, therefore, with very high accuracy, if the magnet be made to oscillate only through a small angle, the value for sin 6 in the above expression for the moment of the couple acting on the magnet, which thus becomes m h ^ ; so that, if K be the moment of inertia of the magnet about the ver- tical axis, the time of a small oscillation t is given by the equation T = 2r>^(j^). G G 2 452 Practical PhyHcs, fCn. XVn. T can be observed experimentally, and hence we get an equation to find m h, viz. If we have in addition a relation which gives the ratio of m/h from the two we can find m and h. Such a relation has been obtained above (p. 450), and with the notation there employed we have ?? = i/^tan4. We shall discuss the experimental details shortly. Magnetic Induction, There are some substances in which the action of mag- netic forces produces a magnetic state which lasts only as long as the magnetic forces are acting. Such substances, of which iron is the most marked example, become them- selves temporary magnets when placed in a magnetic field. They are said to be magnetised by induction. They lose nearly all their magnetic property when the magnetising forces cease to act. In most specimens of iron a certain amount of this remains as permanent magnetism after the cessation of the magnetising forces. In very soft iron the amount is very small ; in steel, on the other hand, the greater portion remains permanently. We shall call such substances magnetic. The attraction between a magnet and a magnetic sub- stance is due to this induction. Wherever a line of force from a magnet enters a magnetic substance it produces by its action a south pole. Where it leaves the substance it produces a north pole. Thus, if a magnetic body be brought near a north pole, those portions of the surface of the body which are turned towards the pole become endued generally with south Ch. XVIL § 69.1 Magnetism. 453 • polai properties ; those parts of the surface which are away from the north pole acquire north polar properties. An attraction is set up between the north pole of the magnet and the south polar side of the induced magnet, a repulsion of weaker amount between the north pole and the north polar Bide, so that on the whole the magnetic body is attracted to the north pole. This may even be the case sometimes when the magnetic body is itself a somewhat weak magnet, with its north pole turned to the given north pole. These two north poles would naturally repel each other ; but, under the circumstances, the given pole will induce south polar properties in the north end of the weak magnet, and this south polarity may be greater than the original north polarity of the magnet, so that the two, the given north pole and the north end of the given magnet, may actually attract each other. 69. Experiments with Magnets. {a) To magnetise a Steel Bar. We shall suppose the magnet to be a piece of steel bar about 10 cm. in length and 0*5 cm. in diameter, which has been tempered to a straw colour. The section of the bar should be either circular or rectangular. We proceed first to shew how to determine if the bar be already a magnet. We may employ either of two methods. Take another delicately-suspended magnet — a well-made compass needle will do — but if great delicacy be required, a very small light magnet suspended by a silk fibre. A small mirror is attached to the magnet, and a beam of light, which is allowed to fall on it, is reflected on to a screen ; the motions of the magnet are indicated by those of the spot of light on the screen, as in the Thomson reflecting galvano- meter. Bring the bar into the neighbourhood of the sus- pended magnet, placing it with its axis east and west and its length directed towards the centre of the magnet, at a dis- tance of about 25 cm. away. Then, if n s be the suspended 4S4 Practical Physics. [Ch- XVII. § 69. magnet, n' s' the bar, and if n' be a north end, s' a south end, N s wiU be deflected as in fig. 50 (i). On reversing Fic sa n' s' so as to bring it into position (2), N s will be deflected in the opposite ^ direction. If the action N' S' N/between the two be too small to produce a visible permanent deflexion of the magnet n s, yet, by con- tinually reversing the bar at intervals equal to the time of oscillation of the needle, the effects may be magnified, and a swing of considerable amplitude given to the latter. The swing can be gradually destroyed by presenting the reverse poles in a similar way. This is a most delicate method of detecting tiie mag- netism of a bar, and there are few pieces of steel which will not shew some traces of magnetic action when treated thus The following is the second method. Twist a piece of copper wire to form a stirrup (fig. 51) in which the magnet Fig. 51. can be hung, and suspend it under a bell- jar by a silk fibre, which may either pass through a hole at the top of the jar and be securedabove, or be fixed to the jar with wax or ^^^J>-^ cement If the magnet to be used /^^ be rectangular in section, the stirrup should be made so that one pair of faces may be horizontal, the other vertical when swinging. For very delicate experiments this fibre must be fireed from torsion. To do this take a bar of brass, or other non-magnetic material, of the same weight as the magnet, and hang it in Ch. XVII. §69.] Magnetism, 455 tiie stimip. The fibre will untwist or twist, as the case may be, and the bar turn round, first in one direction then in the other. After a time it will come to rest The fibre is then hanging without torsion. Now remove the torsion-bar and replace it by the magnetic bar which is to be experimented on, without introducing any twist into the fibre. As the stimip will be frequently used again for suspend- ing the magnet, make a mark on the latter so that it can always be replaced in the same position on the stimip. If now the bar is at all magnetised, it will, when left to swing freely, take up a position of equilibrium with its north end pointing to the north, and when displaced from that position, will return to it again after a number of vibrations about it This method would be even more delicate than the last, except that the torsion of the fibre might sometimes make it appear that the bar is magnetised when it is really not sa Having satisfied yourself that the bar is only feebly magnetised, proceed to magnetise it more strongly. This can be done by stroking it with another magnet, using the method of divided touch, or by the use of an electric current In the method of divided touch the bar is placed on two magnets NiSi, N2S2, Fig. 52 ; two other magnets are held as in the figure NjSj ^ and N4S4. They are then drawn outwards from the centre slowly and regularly, from the position shewn in the figure, in which they are nearly in contact, to the ends. The operation is repeated several times, stroking alwajrs from the centre to the ends. Then the bar to be magnetised is turned over top to bottom and again stroked. It will be found to be a magnet with its north pole n 456 Practical Physics. [Ch. xvn. § 69. over Si and its south pole s over n^ In all cases the two ends of the bar rest on opposite poles, and the poles above, which are used for stroking, are of the same name as those below, on which the bar rests. The two magnets used for stroking should have about the same strength. If an electric current be used, the bar may be mag- netised either by drawing it backwards and forwards across the poles of an electro-magnet, or by placing it inside of a long coil of thick insulated wire, such as is used for the coils of an electro-magnet, and allowing a powerful current to pass through the wire It will be much more strongly magnetised if it be put into the coil when hot and allowed to cool rapidly with the current circulating round it. To deprive a steel bar entirely of its magnetism is a difficult matter. The best plan is to heat it to a red heat and allow it to cool gradually^ with its axis pointing east and west. If it be placed north and south, it will be found that the mag- netic action of the earth is sufficient to re-magnetise the bar. {b) To compare the Magnetic Moment of the same Magnet after different Methods of Treatment^ or of two different Magnets, (i) Suspend the magnet in its stirrup under the bell jar, as in fig. 51, and when it i^in equilibrium make a mark on the glass opposite to one end. Displace the magnet slightly from this position, and count the number of times the end crosses the mark in a known interval of time,' say one minute — a longer interval will be better if the magnet continue swinging. Divide this number by the number of seconds in the interval, 60 in the case supposed, the result is the number of transits in one second. Call this n. There will be two transits to each complete oscilla- tion, for the period of an oscillation is the interval between two consecutive passages of the needle through the resting point in the same direction^ and all transits, both right to left * The times of crossinf; the mark must be counted o, i, 2, . . . n. Ch. xvil § 69.] Magnetism. 457 and left to right, have been taken; \n is therefore the number of complete oscillations in one second, and the periodic time is found by dividing one second by the number of oscillations in one second. Hence, t being the periodic time, But we have shewn (p. 452) that MH = 4fl^K/T*. Hence M H = IT* «* K and Now K depends only on the form and mass of the mag- net, which are not altered by magnetisation ; h is the strength of the field in which it hangs, which is also constant ; so that if M], M2, &a be the magnetic moments after different treatments, »|, n^ &c. the corresponding number of transits per second, M| : M2 = «i^ : «j*, &c We thus find the ratio of M| to m^. (2) We can do this in another way as follows : — Take a compass needle, a b (fig. 53) provided with a divided circle, by means of which its direction can be deter- mined, and note its position of equilibrium. Place the magnet at some distance from the compass needle, with its end point- ing towards the centre of the needle and its centre east or west of that of the needle. Instead of a compass needle we may use a small magnet and mirror, with a beam of light reflected on to a scale, as already described (p. 453)- The centre of the magnet should be from 40 to 50 cm. from the needle. The needle will be deflected from its position of equilibrium. Let the deflection observed be 6^1 ; reverse the magnet so that its north pole comes into the position 458 Practical Physics. [Ch. XVIL § 6^ fonnerly occupied by the south pole, and via versd. TTie needle will be deflected in the opposite direction (fig. 53 [2] ). Let the deflection be O^. If the magnet had been uniformly magnetised and exactly reversed we should find that Qi and 63 were the same. Let the mean of the two values be 9 ; so Fig. 53. 8 w \ — >t— 1 K 8 (3) that is the deflection produced on a magnetic needle by a bar magnet of moment m when the line joining the centres of the two is east and west, and is in the same straight line as the axis of the bar magnet But under these circum- stances we have shewn (p. 450) that, if r be the distance between their centres, M=^Hr'tan6. If another magnet of moment m' be substituted for the first, and a deflection d' be observed, the distance between the centres being still r, we have M'=4HHtantf'. Ch. XVII. § 69.] Magnetism. 459 Hence M : M'=tan*^ : tand'. We can thus compare the moments of the same magnet ynder different conditions, or of two different magnets. {c) To compare the Strengths of different Magnetic Fields 0f approximately Uniform Intensity, Let Hi be the strength of the first field, let a magnet swing in it, and let the number of transits per second ob- served as in ip) be if 1, then we have, m being the magnetic moment, Hi=ir*«i*K/M. Now let the magnet swing in the second field, strength Hs, and let n^ be the number of transits per second Then H3=ir^«a* k/m. Hence H| • Hjssllj , l?2 • To realise the conditions of this experiment surround the magnet hanging as in (a) with a soft-iron cylinder of considerable radius in comparison with the length of the magnet The cylinder should be pierced with holes, through which the magnet may be viewed, and the number of transits per second counted in the manner already described (p. 456). The magnetic field within the iron cylinder is thus com- pared with that which the earth produces when tfie cylinder is removed. (^ To measure the Magnetic Moment of a Magnet and the Strength of the Field in which it hangs. For this we have only to combine the results of the observations in (^), and determine the moment of inertia of the magnet about the axis of rotation. Thus, weigh the magnet and let its mass be m grammes ; measure its length with a rule, the calipers, or the beam compass, as may be 460 Practical Physics. [Ch. XVii. § 69. most convenient ; let it be / cm. Determine, by means of the screw gauge, its diameter if it be a circular cylinder, let it be ^ cm. ; or if it be rectangular in shape, the length of that side of the rectangle which is horizonta^ when it is swinging let this be a cm. Then it can be shewn, by the use of the integral calculus, that in the first case, if the section be circular, and in the second, if it be rectangular, 7« + a^' — C-^" >• Thus K can be determined in either case, supposing the stirrup to be so light in comparison with the magnet that its effect may be neglected. If K cannot be found by direct measurement, we must have recourse to the methods of observation described in §D. Thus, K being determined, we know all the quantities involved in the two equations of (p\ with the exception of M and H. The two equations are M H=ir^«*K, -=iHtan6; H ^ ' and from these we obtain by multiplication, M«=:i ir»»*Kr3 tan 6; whence M=7r«r\/ (^Kr tan 6), and by division, j_2w^«*K r«tanO' ' Rotith*s Ri^id Dynamics^ chapter L See also above, pi 167. Cw.XVll. § 69.] Magnetism. / '\ \^S^\^ \ or . i T This is the method actually employed in many unifilar magnetometers, to determine the horizontal intensity of the earth's magnetic force, the only difference consisting in the very delicate arrangements for the accurate determination of the quantities to be measured {e) To detennine the Magnetic Moment of a Magnet of any shape. The method just given involves the measurement of r, the distance between the centre of the magnet and that of the compass needle, and the assumption that this distance is great compared with the dimensions of the magnets, so that they may be treated as solenoidaL In practice these two conditions may not be possible. We might, for example, require to find the magnetic moment about a diameter of a large steel sphere magnetised in any manner. Now the first equation we have used, viz., M H = IT* «^ K, is true for any magnet, provided only that the amplitude of the oscillation is small, and may be applied to Uie case in point To find, then, the value of m, determine h as in (^, using magnets of a suitable form and size. Suspend the given magnet so that it can oscillate about a suitable axis, and determine r either by calculation from its dimensions, or by observations as in § 23 ; count also «, the number of transits per second of any fixed point on the magnet across some fixed marL The formula will then give us m. (/) To determine the Direction of the Earth's Horizontal Force. Consider a magnet, e.g. a magnetised steel disc, free to turn about a vertical axis, which can be inverted on this axis, so that on inversion the side which was the top comes to the 462 Practical Physics. [Ch. XVII. § 69. bottom, and viu versd. Then we have seen (p. 434) that a certain straight line in the body will set itself in a certain direction, namely, that of the earth's horizontal force. We wish to determine this direction. It may of course be found approximately by the use of a compass needle. Find it thus and make two marks on the magnet such that the line joining them is approximately in the required direction, and at the same time is horizontal Let a, b (fig. 54) be the two Fig. 54. marks, o the point in the axis round which the magnet turns which is in the same horizontal plane as a b, and o h the re- quired direction. Take the magnet off its support, and turn it over top to bottom through 180** ; replacing it, we will suppose, in such a manner on the support that the point o is brought back into its former position. When the magnet again comes to rest, the line in the magnet which originally coincided with o H will clearly do so again ; the effect of the change might have been attained by keeping this line fixed and tinning the magnet about it through 180''. Hence, dearly if a' b' be the new position of a b, a b and a' b' meet on o h at k, say, and are equally inclined to it But A B, a' b' being visible marks on the material of the magnet, the directions of these two lines can be identified : the line which bisects them is the direction required, and is thus readily determined. Moreover, it is not necessary that the point o should, when the magnet is turned round, be brought exactly into its old position. The line o h will in any case after the reversal remain parallel to itselfi and a' b' will represent not the new position of a b, but its projection on the horizontal plane o a a The plane of the magnetic meridian will be a vertical plane bisecting the angle between the vertical planes Cm. XVII. § 69.] Magnetism. 463 through the old and new positions of any line a b fixed io the magnet The experiment then in its simplest form may be performed as follows : — Fasten a sheet of white paper down on to the table, and suspend over it a magnet of any shape whatever, hanging freely in a stirrup, as already described, by a fibre which has been carefully fireed from torsion (p. 454). The magnet should be as dose down to the paper as is possible. Make two marks on the magnet, one at each end, and tooking vertically down on it, make two dots on the paper with a fine-pointed pencil, or some other point, exactly under the two marks ; join these two dots by a straight line. Reverse the magnet in its stirrup, turning the top to the bottom, and let it again come to rest Make two dots as before on the paper vertically below the new positions of the marks, and join these two. The line bisecting the angle between the two lines thus drawn on the paper gives the direction of the horizontal component of the earth's force. In performing the experiment thus, serious error is intro- duced if the observer's eye be not held vertically over the magnet in each case. This is best ensured by placing a piece of plane mirror on the table below the magnet, leaving the part of the paper which is just below the mark un- covered, and placing the eye at some distance away, and in such a position that the image of the magnet, formed by reflection in the mirror, is exactly covered by the magnet itself; then if the dot be made on the paper in such a manner as to appear to the observer to be covered by the mark on the magnet, it is vertically below that mark. If the position of true geographical north at the place of observation be known, we can obtain the angle between the true north and the magnetic north from this experiment This angle is known as the magnetic declination. The declinometer, or apparatus used to measure the declination, is constructed on exactly the same principles as those made use of in the foregoing experiment more 464 Practical Physics. {Ch. xvn. § 69. delicate means being adopted to determine the position of the two marks on the magnet with reference to some fixed direction. For an account of these more delicate methods, see Maxwell's * Electricity and Magnetism,' vol ii. part \u» chap, vii., and Chrystal, * Ency. Brit,* article Magnetism. {£). Experiments on Two Magnets, Comparison oj Magnetic Moments. The magnetic moment of a magnet is measured by the maximum couple which the magnet can experience when placed in a field of magnetic force of unit intiensity. If we have a series of two or more magnets rigidly connected together, the magnetic moment of the system will be found by combining the moments of the parts according to the law of the composition of couples — i.e. according to the parallelogram law. Thus, if we have two magnets care- fully magnetised along the axis of figure, whose moments are m and m\ and place them respectively — 1. With their axes parallel and their poles in the same direction; 2. With their axes parallel and their poles in opposite directions; 3. With their axes at right angles ; And if Ml, Mj, Mabe the magnetic moments of the three combinations, respectively, then we have .-. 2M8> = (»i+iw')H (»!-;«')» = Mi«+Ma^ Now let the magnets be rigidly connected together in these three positions in turn, so that the centre of one is vertically below that of the other, and let the times Tj, Tj, Tj of their oscillations about a vertical axis be observed. The magnets may most easily be so fixed in the follow- ing manner : — A B (fig. xxxv) is a small rectangular block suspended by a tine silk fibre attached to a hook at the centre of one bat. Two parallel holes are bored through one pair of ver- Ch. XVII. § 69. J Magnetism, 465 tical £sices, and by inserting the magnets in these they can be placed in the positions i or 2. A third hole is bored at right angles to the former through the other pair of vertical FlC XXJtT. ^^LL7\ nc N^ 39 3S« faces, and by inserting one of the magnets through this the two can be put in position 3. Now the moment of inertia of the system about the vertical fibre is the same for all three positions. Let this moment be k, and let h be the strength of the earth's hori- zontal field. Then we have -, Mo= I H 2 t,«h' — • Mjk^ — '3 T3*H T2* M Tj — Ti /> We thus find the ratio of the magnetic moments without knowing the moments of inertia of the magnets. Again, when the magnets are in posi- tion 3, let n Syti' sf (fig. xxxvi) be traces of the axes of the two, and let n o s be the north and south line marked on paper below the system. From p, any point on o N, draw p q perpendicular to o n. Then, since the magnets are in equilibrium, and the forces acting H U 466 Practical Physics. [Ch. XVIl. § 60. on their poles are parallel to s n, we have by the parallelo- gram law, Q p tn' Ti* — T|*' This can be verified experimentally by construction. Also, since 2 Ma^ssMi^+MjS we have T3* T|* Tj* This formula can be verified by experiment Care must be taken in the construction and in measur- ing the times of swing in order to obtain accuracy in these last two results. Experiments, (a) Determine if the given bar of steel is magnetised. Mag- netise it (d) Compare the moment of the given magnet after mag- netisation (i) by stroking, (2) by the use of an electro-magnet (c) Compare the strength of the magnetic field within a soft-iron cylindrical screen with the normal strength oi the earth's field. (d) Determine the moment of the given bar magnet and the horizontal intensity of the earth's magnetic force. (e) Determine the moment of the g^ven magnetic mass aboot the given axis, using the known value of the earth's horizontal force. (/) Lay down on the table the direction of the magnetic meridian. Enter results thus : — (a) Effect on suspended magnet only visible after five or six reversals of position, isochronous with the time of swing, (fi) Obierved vatoes of iti Observed Yalues of n^ ngS -144 •104 '148 •loi '140 Mean *ioi Mean *I44 MjM,-(ioi)«/(i44)«--492. Cii. XVII. § 69.] Magnetism. 467 {c) Values of n within the cylinder, using the same magnet after the last magnetisation. •070 x)68 Mean '070 Strength of field within : strength without - ('o/o)* : {'144)*. {d) Using the last observations in {b) /f--i44 K (calculated from dimensions)- 379*9 gm.(cm.)' r->40cm. ^-4^30' Whence H» -176 C.G.S. units M«-442'6C.G.S. units. {e) A sphere of radius 2*5 cm. experimented with. Mass 500 gm. Kai25ogm.(cm.)' H-*I76 C.G.S. units W--0273 M » 52*6 C.G.S. units. (/) Shew on a sheet of paper lines drawn parallel to the edge of the table and to the direction of the horizontal com- ponent of the earth's magnetic force respectively. {g) Compare the magnetic moment of the two given magnets, and verify the result that 2M,' » u^ + M,'. S. Comparison of Gravitational and Magnetio Forces. The force with which an ordinary bar magnet attracts a piece of soft iron varies very rapidly with change in the distance of the iron from the pole of the magnet. The following experiment illustrates this point. A small iron sphere, about *5 cm. in diameter, is sus- pended from the ceiling by a long fine thread, so as to be a few centimetres above a table. Beneath it is placed a scale of centimetres, on which stands a vertical piece of glass. H H a 468 Practical Physics. [Ch. xvn. § s. The glass is mounted in such a way that its position on the scale can be easily determined. The reading at the point underneath the sphere when the thread hangs vertically is observed. Place the glass at right angles to the scale, and at some few centimetres away from this point. On bringing a magnet near on the side of the glass removed from the sphere, the sphere is attracted and moves up against the glass. Gradually withdraw the magnet, keeping it at the same level as the sphere, until the sphere just leaves the glass, and observe its position on the scale when this occurs. It will be Fks xxxvU. c CENTIMETRE SCALE found that this position can be determined with considerable accuracy. Let w be the weight of the sphere, / the length of the string. In fig. xxxvii, let b be the point on the scale vertically under c the point of suspension of the sphere ; let A be the centre of the sphere when just leaving the glass. Let A L be horizontal, and equal to y cm. Let s n be the magnet, and a n = x cm. The distances x and y are given by the observations on the scale. Let f be the horizontal com- ponent at A of the force due to the magnet in the direction Ch. XVII. § S.] Magnetism. 469 of the scale Then the sphere is in equUibrium under its weight, the tension of the string, and the force due to the magnet. Hence the component of the tension in the direction A N is equal to f. If, moreover, the axis of the magnet N s is at the same level as a, the magnet exerts no vertical force on the sphere, and the vertical component of the tension is equal to the weight of the sphere ; but since the string is very long (12 feet), the vertical component of the tension is equal very approximately to the whole tension, and thus we get r=w^ = w^ AC / Set the glass plate so that when the sphere is in contact with it its centre may be at distances of i, 2, 3, . . . cm. respectively from b, and determine the corresponding values of Xy :ci, *2i • • • Then plot a curve, taking the values of x as abscissae and the corresponding values of y^jny^ . • . as ordinates. The curve should be found to take the form given by the equation ^ x jc* = c, where c is a constant for reasons which are given in the foot-note.* > If H is the strength of the magnetic field at A due to the magnet, and a the radius of the sphere, k the magnetic susceptibility, and if 8 H represents the rate of change of H per centimetre increase of jr, the distance A N, then it can be shewn that If the force F be due to the action of a long bar magnet, so long that we may without serious error neglect the efifect of the pole 8 com* pared with that of N, then we have H » ^ and from this jr Thus the force F acting towards N is 8 ^km\ I 3 I + Jir>t i*' ind since F- Wj' '/, we have y^cjx^, where c is constant 470 Practical Physics. [Ch. XVII. § & The value of c may be found by taking the values of X dxAy corresponding to some point p on the curve, and substituting them in the equation; then by drawing the curve y = cr*, and comparing it with the result of the experiment, or by calculating the values oi y x s^ for the observed points, we may verify the result Experiment, — ^Verify the relation j' x jr* « C, in the circum- stances described above, and compare the magnetic force upon the iron sphere when its centre is 2 cm. from the end of th# bar magnet with the weight of the sphere. T. Oauss's Terifloation of fhe Law of Magnetie Force. We have seen already (p. 450) that if the law of force between two magnetic poles be that of the inverse square, and if be the angle through which a magnet is deflected from the meridian by a second magnet of moment m at a distance r in the ' end-on ' position, then H tan = -3-. Wliile if \^ be the deflexion due to the same magnet in the • broadside-on ' position, then H tan i// = -,. r^ These results can be verified by the apparatus referred to in § 69 (b), fig. 53. For if we observe the values of ^ and \// corresponding to different values of r, we can shew that H tan = constant = 2 H tan «/^, If we make the more general assumption that the force between two poles m^ m' is mm'jf^^ then we can find the value of the magnetic potential and the magnetic force Ch. XVII. § T.J Magnetism. 47 1 at any point by the same method as we have applied to the simpler case. We shall find M cos B Vs= R = T = n M cos 6 M sin 6 . '• while the equations giving the ratio of m to H become H tan = pj^,, and Htani// = p^^. Hence by observing and i/' we can find n} There are various ways in which we can carry out the experiment ; one has been already described. The following is one which employs a modification of the ordinary method of reading a galvanometer mirror. The deflected magnet n j, which should be very small, is attached to the back of a mirror. This mirror is sus- pended by a fine silk fibre, the point of suspension being vertically above the point o (fig. xxxviii) ; p q is a wooden stand, pivoted so as to turn about a vertical axis through o, and the support carrying the mirror is attached to p q. By this means the mirror always occupies the same position relative to the support carrying the fibre, and errors due to the torsion of the fibre are eliminated. At p and Q are two vertical pins, equidistant from o, the top of the * In the above we have neglected terms depending on 1*1 r^^ zl being the length of the magnet If these are included, then it can be shewn that Htan^.i!:^/i + (!Ltiii!i±l)4% ...|. Thus «?i| - „ { , + (»±1L('L!J) ^ + ...]. tan^ I 6 r* j 472 Practical Physics. [Cii. xviL f T pins being at a greater height above the board than the mirror. The mirror and magnet are enclosed in a wooden or brass case, with glass windows back and front, through which the pins can be seen ; a mica vane is attached to the back of the mirror to damp the oscillations. The whole is mounted on a drawing-board, carrying a sheet of paper, on which a circle of about 20 cm. radius, with o as centre, is drawn. A horizontal scale, a b, divided to millimetres, is adjusted, as described below, to lie in the magnetic meridian, and fixed to the board On looking at the mirror an image of the pin p can be Fig. xxxvuL seen, and by turning the board round o carefully this image can be made to coincide with q. In this case the' line pq is normal to the mirror, and, therefore, if there are no other magnets near, points east and west Draw the east and west line, EW, on the paper, and throughodraw no s perpendicular to it Adjust the scale a b to be perpendicul^ to e w ; the scale then lies in the magnetic meridian. Note the point w in which the east and west line cuts the scale. This is most readily done by holding a piece of fine wire vertically in a small clip, and moving it until the wire, the pin p, and the image of p in the mirror appear in one line ; or it may be Ch. xvu. ^ T.] Magnetism. 473 done by having a pointer attached to the stand q p, the direction of the pointer being that of q p produced. Now place the disturbing magnet with its centre on the circle at e and its north pole pointing east, so that it is in the * end-on ' position ; the mirror will be deflected. Turn the stand p q until the line p Q is again normal to the mirror, and read the position Kj of the pointer on the scale. Reverse the position of the deflecting magnet at e so that the south pole may point east The mirror will be deflected to the other side of the meridian, and another position (Kj) found for k. If we call the deflexions 0i and ^2, and the corresponding distances measured on the scale c^^ c^ we have tan 01 = — : : tan 0, = — ?. ' ow '^ ow Thus the distances Cx^c^ are respectively proportional to tan 01 and tan 03. If the deflecting magnet is perfectly symmetrical, the two distances will be equal. Now place the magnet with its centre at w, and observe again ; let the distances be r/, c^'. Take the mean of the four ^i, c^, ^1', ^2' ; let it be x. It will correspond to a value of tan 0, corrected for want of symmetry in the deflecting niagnet, and for the fact that the deflected magnet may not be exactly at the centre o^ the circle n e s w. Move the deflecting magnet, still with its axis pointing east an(} west, until its centre is at s, and afterwards at N (it is then in the ' broadside-on * position), and observe as before the four distances, d^y d^ ^/, d^ ; let the mean of these be^. Then y is proportional to tan ^^ the corrected deflexion in the * broadside-on * position ; thus j^ tan)// L 6 r^J From this equation n can be found. If the experiments are conducted with care, we obtain « = 2 very approximately 474 Practical Physics. [Ch. XVIL § T. as the result To solve the equation for «, we may fiist omit the terms involving /'/r®, which will be smalL We thus get an approximate value n^. Then substitute this value in the small terms, and we have To obtain an estimate of the value of the correcting term, we may remember that n^ is nearly 2 ; thus the value of the term in /^/r* is 7/^/2 r*. Suppose / = 2 cm., so that the magnet is 4 cm. long, and r= 20cm., then 'jfPJzr^ = 7/200 = 1/30 approximately. By making observations in a similar manner with the deflecting magnet at different distances from o, we (^n verify the fact that tan ^ is inversely proportional to r^. These experiments were first carried out by Gauss. He found that, provided //r were less than J, the results of his own observations were represented by the formulae tan^ = '086870 r-*— '002185/—*, tan \// = -043435 r-« + '002449 ^t which afford a double verification of the law. Experiment, — Verify the law of the inverse square is Gauss method. Enter the results thus : — Value of r . • . 4*56 n y ... 2-20 / - 3. r » 2a Approximate value of /i « 2*08 Corrected value of « « i'99 IT. ifagnetic Induotion due to fhe Earth. A piece of sofl iron placed in a magnetic field becomes magnetised by induction. If the intensity of the field be small, such as that due to the earth, the magnetic moment nduced by it in the iron will be proportional to the com- Ch. XVII. § U.) Magnetism. 475 ponent of the intensity of the field in the direction of the magnetic axis of the bar. A bar of soft iron may be thus magnetised by induction, and by measuring the strength of either pole of the bar we may obtain a measure of the strength of the inducing field. Thus, take a rod of soft iron about i metre long and I centimetre in diameter. Hold it in a vertical position, and hit it three or four sharp blows with a hammer, or allow one end to fall vertically on to a fiag-stone from about 25 cm. three or four times. The rod will be magne- tised along its length, under the action of v, the vertical component of the earth's magnetic force, and the strength of each pole will be proportional to v, and may be written \ v. Since the rod is very thin, the effect of the horizontal force in magnetising it is negligible. Now bring the rod careftilly, still holding it vertical^ until the lower end (the north pole) is in some definite position with regard to a compass needle — e.g. let it be at the same level as the needle, and 10 cm. to the east of its centre. Call the distance between the two r, and let <p be the deflexion of the compass. Then, since the south pole of the bar is so £u* off, the magnetic force at each pole of the compass needle is Xv/H, and if the compass needle is small the forces on the two poles are nearly parallel, so that Xv = r* H tan (^, H being the horizontal component of the earth's magne- tism. Now place the bar with its axis horizontal and north and south, and magnetise it by striking it as before ; the strength of the poles will in this case be Xh, and if the bar be moved carefully, being kept horizontal, and with its axis north and south all the time, until the north pole comes into the same position as before, and the deflexion now observed in the compass needle be 1//, then Xh = r*H tan 1//. 476 Practical Physics. [Ch. XVIL | 0. Now, if I be llie magnetic dip, H tan\j^ Thus the dip can be found from observations of ^ and i^. To obtain an accurate result the experiment must be repeated, care being taken to strike the bar sufficiently in each position to ensure its receiving the maximum amount of magnetisation which the horizontal and vertical forces; respectively, are capable of inducing. Experiment. — Determine the dip* by observations on the magnetism induced in a rod by the horizontal and vertical com- ponents of the earth's magnetic field. 70. Exploration of the Magnetic Field due to a given Magnetic Distribution. Place a bar magnet on a large sheet of paper on a table In the neighbourhood of the magnet there will be a field of magnetic force due to the joint action of the earth and the bar magnet, and if a small compass needle be placed with its centre at any point of the field, the direction of the needle, when in equilibrium, will indicate, very approxi- mately indeed, the direction of the line of magnetic force which passes through its centre. Draw a line on the paper round the bar magnet at a distance of 2 or 3 cm. from it, and mark off points along this line at intervals of 2 cm. Take a small compass needle and lay it so that its centrt is above the first of the points so marked ; it will then set itself in the direction tangential to the line of force which passes through the point Make marks on the paper exactly opposite to the points at which the ends of the ' The student should notice that this experiment merely illustnitei the proportionality between the small magnetising forces and tbe corresponding magnetisation. It is not a standard method of deter- mining the dip. Fig. 55- T~i — n — I — I 1 I Ch. XVII. § 7aj Magnetism. 4;; compass rest, and as close to them as possible. Let a b (fig. 55) be the ends of the compass. Move the compass on in the direction in which it points, and place it so that the end a comes ex- actly opposite the mark against the old position of B, while the end b moves /I/ on to position c, so that d*'-^ 'c^^' ^ B c is the new position of the compass. Make a mark opposite the point c in its new position. Again move the compass on until the end at b comes into the position c, and so on. A series of points will thus be drawn on the paper, and a line which joins them all will very nearly coincide with a line of force due to the given distribution. The line of force can thus be traced until it either cuts the line drawn round the magnet or goes off the paper. Repeat the operations, starting from the second of the points on the line drawn round the bar magnet, and then from the third, and so on, until the lines of force for all the points are drawn, thus giving a com- plete map of the directions of the lines of force due to the combination.^ It is convenient to have the compass needle mounted, as is often done for trinkets, between two pieces of glass. The dots on the paper can then be seen through the glass, and the compass set so that the end of the needle may be accurately over the dot. If, further, the compass have a * However the bar magnet be placed, there will generally be found two points in the field at which the resultant force is zero. These points can be very accurately identified by carefully drawing the lines of force in their neighbourhood. When they have been determined their distances from the poles of the bar magnet can be measured by a scale ; the angles between the lines joining one of the points of zero force with the poles can be determined, and from these observations an estimate can be made of the strength of either pole of the bar magnet in terms of the strength of the earth's field. The positions of the poles are very well indicated by the convergence of the lines of force. 473 Practical Physics, [Ch. xvil. § 7a non-magnetic ann fixed at right angles to the needle, tber the direction of this arm gives the direction of the equi- potential surface at the point, and by making dots under Fic. xxxix. the ends of this arm, and working with it in the same way as with the needle itself, we can draw the equipotential surfaces. Fig. xxxix is a set of such lines drawn in this way. Experiment — Draw a map of the directions of the lines of force due to the combined action of the earth and the given bar magnet Ch. XVII. § v.] Magnetism, 479 V. Magnetic Induction in Iron. The magnetic force at a point has been defined as the force on a unit pole placed at that point Now if the point be in the middle of a magnet, such as a mass of iron or steel in a magnetic field, we must suppose a small cavity removed in order to place the unit pole there. We can shew that the force on the pole depends on the shape of the cavity (see Ewing, ' Magnetic Induction in Iron and othei Metals,' pp. 1-22), for the magnetic forces induce on the walls of the cavity magnetism, which acts on the pole, and the effect of the magnetism so induced depends on the shape of the cavity. The iron or steel is magnetised by the external field. Let us suppose the cavity takes the form of a long narrow cylinder, with its length along the lines of magnetisa- tion. Then the force on the pole is defined as the magnetic force inside the cavity ; we denote it by H. If, on the other hand, the cavity is a very narrow crevasse at right angles to the direction of magnetisation, then the force on unit pole in such a cavity defines the magnetic induction ; we denote it by B. The ratio of B to H is generally denoted by fi, and is called the permeability. The per- meability is not a constant, but depends on the value of H and on the past history of the iron. When the iron is subject to magnetic force each small element of volume V becomes a magnet ; let us denote the moment of that element by I v^ so that I is the magnetic moment per unit volume of the iron. I is called the intensity of magnetisation. The ratio of I to H is the susceptibility, and is denoted by <. The susceptibility, like the permeability, is not constant, but depends on H and on the past history of the iron. Now we may shew * that B = H+4Jr I =(l+4'r/c)H =/i H by definitiott > * See £win^, loc. Hi, 48o Practical Physics. [Ch. XVIL § ?. The induced magnetisation produces magnetic force inside the magnetised body, which acts in the opposite direction to the magnetising force. The amount of diis induced force depends on the shape and material of the magnetised body. Thus if a long rod be magnetised by a force Ho, one end becomes a north pole, the other a south pole, and within the rod we have, in addition to the force Hq, the opposing force due to the ends. In any calculation, then, the effect of this must be allowed for ; but if we make the length of the rod very long compared with its diameter (say 400 times the diameter), the effect of the ends is negligible except near the ends, and we may treat the problem as though the magnetic force in the rod were the impressed force Ho* Now if a current be allowed to circulate in a long coil of insulated wire wound into the form of a close straight helix, the lines of force inside the helix, except near its ends, are straight lines parallel to the axis of the helix (see Searle,* 'Determination of Currents in Absolute Measure'), and it can be shewn that if y be the current in absolute electro- magnetic measure, and n the number of turns per unit length of the helix, then inside the helix Ho=4 ^ a yf If then a thin rod of soft iron be placed inside the helix, we can subject it to a known magnetising force, and examine in the following way the effects. Place the helix horizontally, with its axis east and west, in such a position that the axis produced passes through the centre of a magnetometer needle. A small mirror, with a magnet at its back, suspended by a silk fibre, and a lamp and scale are arranged in the usual manner. The coil may conveniently be about 50 cm. in length and I cm. in diameter, wound with two or more layers of in- sulated wire. The ends of the wires are connected through a tangent * Mr. Searle*s papers appeared in the Electricicm for 1 891. • See Ewing, he* cU. Ch. XVII. § v.] Magnetism. 481 galvanometer g (^%, xl), or a direct-reading ammeter and an adjustable resistance r, to a battery and a reversing key k. On passing a current through the coil c the magneto- meter is affected by the direct action of the coil. This action may be compensated by a permanent magnet. It is better, however, to pass the same current through a second coil d of larger area with a few turns of wire, placed Fig. xL near the magnetometer, this coil can be adjusted so that its effect on the magnetometer is exactly opposite to that of the main coil. Make this adjustment for the largest current which is to be used, and secure the coil c' in position with a clamp. Then the currents in the coils will not affect the magnetometer, and any action which takes place is due to the magnetism induced in the soft iron rod when it is put in. The leading-wires should be kept close together and not moved. (i) To find the Magnetic Moment of a Soft Iron Rod, The rod may be 40 cm. long by 'i cm, thick. See, in the usual way, that the rod is free from permanent magnetism. If not, heat it to a red heat and allow it to cool in an east and west position. Insert it in the helix, and let its centre be r cm. distant from the magnetometer ; let the length of the rod I I 482 Practical Pkysics. [Ch. xvn. § V. be 2 / cm., and let m be the induced magnetic moment, m the strength of either pole, assuming the magnetisation uniform. The rod should be distinctly shorter than the helix. Then the magnetic force in the direction of its axis at distance r from its centre is m m and this is equal to 4mri or to 2Mr_ If be the deflexion of the magnetometer, then this magnetic force is equal to h tan 0. Thus we have M=H tan^V — Z-_i_. 2r In making the observations it is desirable to tap the rod lightly when in position ; this helps the magnetisation. The value of m should be measured for different rods. By taking rods of the same thickness, but of different lengths, we can examine the effect of the ends ; if this effect be inappreciable the values found for m will be proportional to the respective lengths. In order to secure this the ratio diameter to length should not be greater than i/4oa (2) To find the Magnetic Susceptibility. Take a rod in which the effect of the ends is known to be small, and measure its magnetic moment h. Let 2 / be its length, and a the radius of a section which we suppose b circular j then its volume is 2 ^ /a^, and if I is the in- tensity of magnetisation, | is the magnetic moment per unit volume. Ch.XVII §V.] Magnetism, 483 rhus I can be found Since we may neglect the ends of the rod, the magnetic force inside it is H = 4 T ff y, and r, the susceptibility, is the ratio of I to H, M • • tf := _. Htan^(r' — Z')* i6w*tf^r/fiy Let G be the galvanometer constant of the galvanometer used to measure the current (see p. 503), and the de- flexion of the magnet ; then y=-tane: G /. yGtan» (r«-/^)« tan 8 i67r*a*r/«' The same observations give us ^, the permeability, for ^ = I + 4 TT «. Now break the battery circuit The rod Ficzll o < UJ z o I MAGNETIC FORCE will remain magnetised, though to a less extent than before, the amount of residual magnetisation depending largely on the method adopted for breaking the current. Measure the 1 1 2 484 Practical Physics, [Ch. XVIL f v. residual moment m in the same way, and calculate the residual susceptibility, viz. by the ratio of the residual magnetisa- tion to the maximum magnetising force. Now free the rod from magnetisation, and repeat the experiment, using a dif- ferent magnetising current Plot the results on a curYe, taking the values of the magnetising force H as abscissae and the corresponding magnetisations as ordinates. The curves wiU have the form shewn in fig. xlL (3) Magnetic Cycles. Hysteresis. The behaviour of iron in a magnetic field can be more completely investigated if the magnetic force be carried through a complete cycle of changes in the following manner. Include in the battery circuit a variable resistance This may consist either of an adjustable wire rheostat, or, better, of a liquid resistance, such as copper sulphate. This may be contained in a rectangular trough. A fixed copper plate dips into one end of the trough, while a second movable plate can be inserted in any other position. The trough, which is shown at R, fig. xl, is tilted, so that the depth of the liquid is much greater near the fixed plate than at the other end, where it only just covers the base. If the movable plate be inserted at this end, a very large resistance is in circuit ; as the plate is moved tows^ds the other end the resistance decreases. The battery circuit should also contain a reversing key. We wish to investigate the magnetisation of the rod as the magnetising force gradually increases from zero up to a maximum, and then decreases again through zero to an equal negative maximum, from which it is again increased through zero up to the same positive maximum as before. The adjustments are made as already described, the movable plate being placed so that the resistance in circuit at starting is very great, and the current made. A series of simultaneous readings of the galvanometer and magneto- Ch. XVII. § v.] Magnetism, 485 meter are then taken, the resistance being gradually de- creased. When the current has reached its maximum value the resistance is again gradually increased and the current reduced to zero ; if the results be plotted it will be found that the descending curve is much less steep than the ascending, and when the current is zero there will be a con- siderable amount of residual magnetism left The battery commutator is then reversed, and the resistance again diminished until the current reaches a maximum negative value. It will be found that during this process the mag- netisation does not at first alter much, but that after the current has attained a not very large negative value there is a sudden large change in the magnetisation from a con- siderable positive amount to an equally large negative value. After this, as the current increases the magnetisation in- 486 Practical Physics. [Ch. XVIL § v. creases, but more gradually. When the current has reached its maximum negative value it is again decreased by in- creasing the resistance, and afterwards, passing through zero, reversed and increased again up to the same positive maximum as before. If the magnetisation curve for this process be drawn, it will be a closed curve, ^ resembling in form that given in figure xlii. Again, it has been shewn that the area measured on a proper scale of the closed cycle is the total energy re- quired to carry unit volume of the iron through the mag- netic changes. This energy is dissipated as heat Moreover, Prof. Ewing has shewn that whenever iron is taken through any cyclic process of magnetising force, the magnetisation changes, but in such a way as always to lag behind the magnetising force ; there is a tendency for the existing state of magnetisation to persist To this tendency he has given the name hysteresis, and it is in consequence of this hysteresis that energy is required to produce a cycle of magnetic changes. Experiments, (i) Determine the magnetic moment of the given pieces of soft iron under a given magnetic force. (2) Find the susceptibility and permeability of soft iron for various values of the magnetising force, and determine also the residual magnetisation when the force is suddenly re- moved. (3) Draw the hysteresis curve for the given specimen of soft iron, and calculate the energy dissipated as heat in carrying it round a complete cycle. Enter in parallel columns the values of H, I, 1^ B, ^ and draw the curve. > For a discussion of the properties of this curre, and the yariatioiii in its form for varioas specimens of iron, see Ewing, Magnetic In- duction in Iron^ &c., chaps, iv. and ▼., from which much of the above is taken. c:h. xviil] Electricity. 487 CHAPTER XVIII. ELECTRICITY — DEFINITIONS AND EXPLANATIONS OF ELECTRICAL TERMS. In the last chapter we explained various terms relating to magnetism. Just as in the neighbourhood of a magnet we have a field of magnetic force, so, too, in the neighbour- hood of an electrified body there is a field of electric force. We proceed to consider certain facts, and to explain some of the terms connected with the theory of electricity, a clear comprehension of which will be necessary in order to understand rightly the experiments which follow. Most bodies can by fiiction, chemical action, or by various other means, be made to exert forces on other bodies which have been similarly treated. The phenomena in ques- tion are classed together as electrical^ and the bodies are said to have been electrified. By experiments with Faraday's ice- pail among others {vide Maxwell's * Elementary Electricity,' p. 16, &c), it has been shewn that these effects can be ac- counted for by supposing the bodies to be charged with certain quantities of one of two opposite kinds of electricity^ called respectively positive and negative, and such that equal quantities of positive and negative electricity completely annihilate each other. An electrified body exerts force on other electrified bodies in its neighbourhood — in other words, produces a field of electrical force — and the force at any point depends on the position of the point, on the form and dimensions of the electrified body, and on the quantity of electricity on the body. By doubling the charge we can double the force. We are thus led to look upon electricity as a quantity which can be measured in terms of a unit of its own kind, and we may speak of the quantity of electricity on a body, in some- irhat the same way as we use the term quantity of magnetism for the strength of a magnetic pole. The magnetic forces 488 Practical Physics. [Ch. XVTIL produced by a magnetic pole are due to a quantity of mag- netism concentrated at the pole. The electrical forces produced by an electrified body are due to a quantity of electricity distributed over the body. By supposing the txidy to become very small while the quantity of electricity on it still remains finite, we may form the idea of an electxified point or a point charged with a given quantity of electricity. With regard to the transmission of electrical properties bodies may be divided into two classes, called respectively conductors and non-conductors. To the latter the name 'dielectric ' is also applied. DsFiNrnoNS of Conductors and Non-conductors. — If a quantity of electricity be communicated to a conductor or conducting body at one point, it distributes itself accord- ing to certain laws over the body ; if, on the other hand, it be communicated to a non-conductor, it remains concentrated at the point where it was first placed. Quantities of electri- city pass freely through the substance of a conductor ; they cannot do so through a non-conductor. Quantities of electricity are of two kinds, having opposite properties, and are called positive and n^;ative respectively. Two bodies each charged n^ith the same kind of electricity repel each other ; two bodies charged with opposite kinds attract each other. To move an electrified body in the field of force due to an electrified system, against the forces of the sys- tem requires work to be done, depending partly on the forces of the system and partly on the quantity of electricity on the body moved. — We shall see shortly how best to define the unit in terms of which to measure Uiat quantity. — Moreover, owing to the action between the electrified body and the rest of the system, alterations will generally be produced in the forces in consequence of the motion. Definition of Resultant Electrical Force. — ^The resultant electrical force at a point is the force which would be exerted on a very small body charged with unit quantity of positive electricity placed at the point, it being supposed Ch. xviii.] Electricity. 489 that the presence of the body does not disturb the electrifi- cation of the rest of the system. Hence if R be the resultant electrical force at a point, and e the number of units of electricity at that point, the force acting on the body thus charged is r ^ If the body so charged be moved by the forces acting on it, work is done. Definition of Electromotive Force. — The work done in moving a unit quantity of positive electricity from one point to another is called the electromotive force between those points. Hence, if the electromotive force (denoted by the symbols E.M.F.), between two points be e, the work done in moving a quantity e of positive electricity from the one point to the other is e e. Electromotive force is sometimes defined as the force which tends to move electricity ; the definition is misleading. The name itself is perhaps ambi- guous, for the electromotive force between two points is not force, but work done in moving a unit of positive electricity; it, therefore, has the dimensions of work divided by electrical quantity (see p. 20). The term electromotive force at a pointy however, is sometimes used as equivalent to the re- sultant electrical force. We shall avoid the term. Suppose that a single body charged with positive electricity is being considered, Uien it is found that the force which this body exerts on any electrified body de- creases very rapidly as the distance between the two bodies is increased, becoming practically insensible when the distance is considerable. We may define as the field of action of an electrified system of bodies that portion of space throughout which the electrical force which arises from the action of those bodies has a sensible value. If a quantity of positive electricity be moved from any point of the field to its boundary by the action of the electrical forces, work is done. Definition of Electrical Potential. — The electrical potential at a point is the work which would be done by the 490 Practical Physics. [Ch. xvm electrical forces of the system in moving a unit quantitj of positive electricity from the point to the boundary of the field, supposing this could be done without disturbing the electrification of the rest of the bodies in the field. We may put this in other words, and say that the electrical potential at a point is the E.M.F. between tfa^ point and the boundary of the field. It is clear from this definition that the potential at aO points of the boundary is zero. The work done by the forces of the S3rstem, in moving a quantity e of positive electricity from a point at potential v to the boundary, is clearly v^ and the work done in moving the same quantity from a point at potential v, to one at potential Vj is ^(Vi— Vj). Hence, it is clear that the E.M.F. between two points is the difference of the potentials of the points. We are thus led to look upon the electric field as divided up by a series of surfaces, over each of which the potential is constant The work done in moving a unit of positive electricity from any point on one of these to any point on another is the same. When two points are at difiierent potentials there is a tendency for positive electricity to flow from the point at the higher to that at the lower potential If the two po^'nts be connected by a conductor, such a flow will take place, and unless a difference of potential is maintained between the two points by some external means, the potential will become equal over the conductor ; for if one part of the conductor be at a higher potential than another, positive electricity immediately flows from that part to the other, decreasing the potential of the one and increasing that of the other until the two become equalised. Now the earth is a conductor, and all points, not too far apart,* which are in metallic connection with the earth are at the same potential ' If the points are far apait, electro-magnetic effects are produced by the action of terrestrial magnetism. Ch. XVIIL] Electricity. 491 It is found convenient in practice to consider this, the potential of the earthy as the zero of potential ; so that on this assumption we should define the potential at a point as the work done in moving a unit of positive elec- tricity from that point to the earth. If the work done in moving a unit of positive electricity from the earth to the boundary of the field be zero, the two definitions are identical ; if this be not the case, the potential at any point measured in accordance with this second definition will be less than its value measured in accordance with the first definition by the work done in moving the unit of positive electricity from the earth to the boundary of the field; but since electrical phenomena depend on difference of potential, it is of no consequence what point of reference we assume as the zero of potential, provided that we do not change it during the measurements. In either case the E.M.F. between two points will be the difference of their potentials. Potential corresponds very closely to level or pressure in hydrostatics. The measure of the level of the water in a dock will depend on the point from which we measure it, e,g, high water- mark, or the level of the dock-sill below high water-mark; but the flow of water from the dock if the gates be opened will depend not on the actual level, but on the difference between the levels within and without the dock, and this will be the same from whatever zero we measure the levels. Various methods have been discovered for maintaining a difference of potential between two points connected by a conductor, and thus producing between those points a con- tinuous flow of electricity ; the most usual are voltaic or galvanic batteries. For the present, then, let us suppose that two points A and B are connected with the poles of a battery, a and b being points on a conductor, and let us further suppose that the pole of the battery connected with a is at a higher potential than that connected with b. The pole connected with A is said tp be the positive pole. A continuous transfer 492 Practical Physics. [Ch. xvnt \ of positive electricity will take place along the conductor from A to B. Such a transfer constitutes an electric current Let p Q (fig. 56) be any cross-section of the conductor between the points a and b, dividing it into two parts. Then it is found that during the same interval the quantity of electricity which in a given time {say one second) flows across the section p q is the same for all positions ofFQf provided only that a and B are on opposite sides of the section. Thus, if in the figure p'q' be a second section, then at each instant the same quantity of electricity crosses p q and p' q' per second. The laws of the flow of electricity in conductors re- semble in this respect those which regulate the flow of an incompressible fluid, such as water, in a tube ; thus, if the conductor were a tube with openings at a and b, and if water were being poured in at a and flowing out at b, Ac tube being kept quite full, then the quantity of water which at any time flows in one second across any section of the tube, such as p q, is the same for all positions of p Q, and as in the case of the water the quantity which flows depends on the difference of pressure between a and b, so with the electricity, the quantity which flows depends on the E.M.F., or difference of potential between the points.' Definition of a Current of Electricitv. — A current of electricity is the quantity of electricity which passes in one second across any section of the conductor in which it is flowing. Thus, if in one second the quantity which crosses any section is the unit quantity, the measure of the current is unity. A unit current is said to flow in a conductor when unit ■ Maxwell's EUnuntary Blectricity, % 64. Ch. xvilL] Electricity. ^ 493 quantity of electricity is transferred across any section in one second. But as yet we have no definition of the unit quantity of electricity. To obtain this, we shall consider certain other properties of an electric current A current flowing in a conductor is found to produce a magnetic field in its neighboiurhood. Magnetic force is exerted by the current, and the pole of a magnet placed near the conductor will be urged by a force definite in direction and amount If the conductor be in the form of a long straight wire, a north magnetic pole would tend to move in a circle round the wire, and the* direction of its motion would be related to the direction of the current in the same way as the direction of rotation is related to that of transla- tion in a right-handed screw. If instead of a magnetic pole we consider a compass needle placed near the wire, the needle will tend to set itself at right angles to the wire, and if we imagine a man to be swimming with the current and looking at the needle, then the north end will be turned towards his left hand. As to the intensity of the force, let us suppose that the length of the wire is / centimetres, and that it is wound into the form of an arc of a circle r centimetres in radius; then when a current of intensity / circulates in the wire, it is found that the magnetic force at the centre is proportional to li\f^ and acts in a direction at right angles to the plane of the circle, and if / be measured in proper units, we may say that the magnetic force is equal * to /i/^. If we call this f, we have Let the length of the wire be one centimetre, and the radius one centimetre, and let us inquire what must be the strength of the current in order that the force on a unit magnetic pole may be one dyne.' " Sec p. 500. • See duip. U. p- 18. 494 Practical Physics. [Ch, XVIII. We have then in the equation Fs= I, /= I, r= 1, and it becomes therefore /=i; that is, the strength of the current is unity, or the cunent required is the unit current Thus, in order that the equation may be true, it b necessary that the unit current should be that current which circulating in a wire of unit length, bent into the form of an arc of a circle of unit radius, exerts unit force on a unit magnetic pole placed at the centre. But we have seen already that the unit current is obtained when unit quantity of electricity crosses any section of the conductor. We have thus arrived at the definition of unit quantity of electricity of which we were in search. This detinition is known as the definition of the electro- magnetic unit of quantity. Definition of CG.S. Electro-Magnetic Unit Quan- tity AND Unit Current. — Consider a wire one centimetre in length bent into an arc of a circle one centimetre in radius. Let such a quantity of electricity flow per second across any section of this wire as would produce on a unit magnetic pole placed at its centre a force of one dyne. This quantity is the electro-magnetic unit of quantity of electricity, and the current produced is the electro-magnetic unit of current. With this definition understood then, we may say that if a current of strength i traverse a wire of length / bent into an arc of a circle of radius r, the force on a magnetic pole of strength m placed at the centre of the circle will htrntifr^ dynes in a direction normal to the circle, and the strength of the magnetic field at the centre is i//r*. The magnetic field will extend throughout the neigh- Ch. XVIIL] Electricity. 495 bourhood of the wire, and the strength of this field at any point can be calculated. Accordingly, a magnet placed in the neighbourhood of the wire is affected by the current, and disturbed from its normal position of equilibrium. It is this last action which is made use of in galvano- meters. Let the wire of length / be bent into the form of a circle of radius r, then we have /= 2irr, and the strength of the field, at the centre of the circle, is 2 IT //r. Morepver, we may treat the field as uniform for a distance fi-om the centre of the circle, which is small com- pared with the radius of the circle. If then we have a magnet of moment m, whose dimensions are small com- pared with the radius of the circle, and if it be placed at the centre of the circle so that its axis makes an angle B with the lines of force due to the circle, and therefore an angle of ^o^ — 6 with the plane of the circle, the moment of the force on it which arises from the magnetic action of the current is 2 ^ m / sin Ojr, If, at the same time, ^ be the angle between the axis of the magnet and the plane of the meridian, the moment of the force due to the horizontal component h of the earth's magnetic force is m h sin ^ ; if the small magnet be sup- ported so as to be able to tiun round a vertical axis, and be in equilibrium under these forces, we must have the equation 2 IT M / sin ^ = M H sin ^9 or . H r sin ^ , 2 ^ sm ^ if then we know the value of h, and can observe the angles iff and (^, and measure the distance r, the above equation gives us the value of /. 496 Practical Physics. [Ch. xvm Two arrangements occur usually in practice. In the first the plane of the coil is made to coincide with the mag- netic meridian ; the lines of force due to the coil are theD<a( right angles to those due to the earth, and fl = 9o«» - <^ Hence sin 6 = cos ^ and we have ._^ Hrtan ^ The instrument is then called a tangent galvanometer. In the second the coil is turned round a vertical axis until the axis of the magnet is in the position of equilibrium in the same plane as the circle ; the hnes of force due to the coil are then at right angles to the axis of the magnet, so that the effect of the current is a maximum, and ^=90^ In these circumstances, therefore, we have, if ^ be the deflection of the magnet, H r sin ^ 2 v The instrument is in this case called a sine-galvanometer. We shall consider further on, the practical forms given to these instruments. Our object at present is to get clear ideas as to an electric current, and the means adopted to measure its strength. The current strength given by the above equation will, using CG.S. units of length, mass, and time, be given in absolute units. Currents, which in these units are repre- sented even by small numbers, are considerably greatei than is convenient for many experiments. For this reason, among others, which will be more apparent further on, it b found advisable to take as tht fractical unit of current^ one- tenth of the CG.S. unit This practical unit is called an ampere. Ca XVIII.] Electricity. 49? Definition of an AMPiRE. — ^A current of one ampere is one-tenth of the CG.S. absolute unit of current Thus, a current expressed in CG.S. units may be reduced to amperes by multiplying by la CHAPTER XIX. experiments on the fundamental properties of electric currents — measurement of electric cur- rent and electromotive force. 71. Absolute Heasure of fha Current in a Wire. The wire in question is bent into the form of a circle, which is placed approximately in the plane of the magnetic meridiaa This is done by using a long magnet mounted as a compass-needle and placing the plane of the wire by eye parallel to the length of this magnet The two ends of the wire are brought as nearly into contact as is possible, and then turned parallel to each other at right angles to the plane of the circle ; they are kept separate by means of a small piece of ebonite, or other insulating material. A small magnet is fixed on to the back of a very light mirror, and suspended, by a short single silk fibre, in a small metal case with a glass face in front of the mirror, just as in a Thomson's mirror galvanometer. The case is only just large enough to allow the mirror to swing freely, so that the air enclosed damps the vibrations rapidly. The case is fixed to an upright stand and rests on levelling screws in such a way that the centre of the magnet can be brought into the centre of the circle. A scale parallel to the plane of the circle is fixed some little distance in firont of the mirror, the level of the scale being very slightly above that of the mirror. Below the scale is a slit, and behind that a lamp, the light from which shines through the slit on to the K K 498 Practical Physics. [Ch, XIX. § 71. mirror, and is reflected by it, throwing a bright spot of light on to the scale, if the scale and lamp be properly adjusted The mirror is usually slighdy concave, and by adjusting the distance between the scale and the mirrofy a distina image of the slit can be formed on the scale, and its position accurately determined. In some cases it is convenient to stretch a thin wire vertically across the middle of the slit, und read the position of its image. If an image cannot be obtained by simply varying the distance, through the mirror not being concave, or from some other defect, a convex lens of suitable focal length may be inserted between the sht and the mirror; by adjusting the lens the image required can be obtained. When there is no current passing through the wire the image should coincide with the division of the scale which is vertically above the slit To determine whether or not the scale is parallel to the mirror, mark two points on the scale near the two ends, and equidistant from the middle point, and measure with a piece of string the dis- tances between each of these two points and a point on the glass face of the mirror-case exactly opposite the centre of the mirror. If these two distances be the same, the scale is rightly adjusted; if they be not, turn the scale, still keeping the image of the slit vertically above the slit, until they become equal Then it is clear that the scak is at right angles to the line which joins its middle point to the mirror, and that this line is also at right angles to the mirror. The scale, therefore, is parallel to the mirror. If now the ends of the wire be connected with the poles of a Daniell's battery, or with some other apparatus which maintains a difference of potential between them, a current will flow in the wire. The magnet and minor wiH be deflected, and the spot of light will move along the scale, coming to rest after a short time in a diflerent position. Note this position, and suppose the distance between it and the original resting-point to be x^ scale divisions — ^it will be convenient when possible to use a scale divided into Ch.XIX. § 71.] Experiments an Electric Currents. 499 centimetres and millimetres. — Reverse the direction of the current in the circuit, either by using a commutator or by actually disconnecting it from the battery, and connecting up in the opposite way. The spot will be deflected in the opposite direction through, let us suppose, x^ scale divisions. If the adjustments were perfect, we ought to find that Xx and x^ were equal ; they will probably differ slightly. Let their mean be x. Then it can be shewn that, if the difference between x^ and x^ be not large, say about 5 scale divisions, when the whole deflexion is from 100 to 200 divisions, we may take x as the true value of the deflexion which would have been produced if the scale and mirror had been per- fectiy adjusted. Let us suppose further that a large number of scale divisions — say 500 — occupies / cm. Then the number of centimetres in x scale divisions is xij $00. Measure the distance between the centre of the mirror and the scale, and let it be a cm. Measure also the diameter of the circle in centimetres, estimating it by taking the mean of measurements made in five different (Erections across the centre. Allow for the thickness of the wire, and so obtain the mean diameter of the core of the circle formed by the wire ; let it be 2 r centimetres. Let bab' (fig. 57) be the scale, a the slit, and b the point at which the image is formed ; let c be the centre of the mirror ; the ray of light fig. 57. has been turned through the angle a c b, and if <^ be the angle through which the mag- net and mirror have moved, then ACB= 2^ for the reflected ray moves through twice the angle which the mirror does (see § 48). Moreover, the distances c a and A b have been observed, and we have a b ssxlj $00, c A =tf. v V 1 500 Practical Physics. [Ch. xdc § 71. Thus 500 a CK From this equation then 2 ^ can be found, using a tabic of tangents, and hence tan ^ by a second application of the table. But the cirde was placed in the magnetic meridian, parallel, therefore, to the magnet, and the force due to the current is consequently at right angles to that due to the earth. We have, therefore, from the last section, if / re- present the current, /=Hr tan ^/2ir. We have shewn in § 69 how h is to be found, and the values of r and tan ^ have just been determined ; the value. of ir is, of course, 3*142, and h may be taken as 'i8a Thus we can measure 1 in C.G.S. absolute units. To find i in amperes we have to multiply the result by 10^ since the CG.S. unit of current contains 10 amperes. The repetition of this experiment with circles of difierenl radii would serve to demonstrate the accuracy of the funda- mental law of the action of an electric current on a magnet The experiment may, by a slight modification, be arranged with the more direct object of verifying the law in the follow- ing manner. Set up two coils concentrically, in the magnetic meridian, with a needle at their common centre. Let the one coil consist of a single tiun of wire and the other of two turns, and let the radius of the second be double that d the first Then on sending the same current through either coil the deflexion of the needle will be found to be the same ; the best way, however, of demonstrating the equalitj is to connect the two coils together so that the sanu cur- rent passes through both, but in opposite directions; the effect on the needle for the two coils respectively being equal and opposite, the needle will remain undefiected. We arc indebted to Professor Poynting, of Birmingham, for the Ch. xnc § 71.] Experiments an Electric Currents. 501 suggestion of this method of verifying the fundamental electro-magnetic law. It should be noticed that the formula for the deflexion does not contam any factor which depends on the magnetism of the suspended needle ; in other words, the deflexion of a galvanometer is independent of the magnetic moment of its needle. This fact may also be experimentally verified by repeating the experiment with diflferent needles and noticing that the deflexion is always the same for the same current Experiment. — Determine the strength of the current from the given battery when flowing through the given circle. Enter results thus : — Observations for diameter, corrected for thickness of the wire— 53 cuL 32*1 cm. 31*9 cm. 33 cm. 32*1 cm. Mean value of r, 16*01 cm. x^ 165 divisions of scale. /» space occupied by 500 divisions - 317 cm. a-607 cm. tan3^--i7a3 tan ^--0855 /•x>3925 CG.S. unit -'3925 ampere. GALVANOMBTERS. The galvanometer already described, as used in the last section, was supposed to consist of a single turn of wire, bent into the form of a circle, with a small magnet hanging at the centre. If, however, we have two tmns of wire round the magnet, and the same current circulates through the two, the force on the magnet is doubled, for each circle producing the same effect, the effect of the two is double that of one ; and if the wire have n turns, the force will be n times that due to a wire with one turn. Thus the force which is produced by a current of strength 502 PracHcal Physics. [Ch. xdl ^ at the centre of a coil of radius r, having n turns of wire, is 2 HIT ijr. But we cannot have n circles each of the same radius, having the same centre ; either the radii of the difierent circles are different, or they have different centres, or both these variations from the theoretical form may occur. In galvanometers ordinarily in use, a groove whose section is usually rectangular is cut on the edge of a disc of wood or brass, and the wire wound in the groove. The wire is covered with silk or other insulating material, and the breadth of the groove parallel to the axis of the disc is such that an exact number of whole turns of the wire lie evenly side by side in it The centre of the magnet is placed in the axis of the disc sjonmetrically with reference to the planes which bound the groove. Several layers of wire are wound on, one above the other, in the groove. We shall call the thickness of a coil, measured from the bottom of the groove outwards along a radius, its depth. Let us suppose diat there are n turns in the galvano- meter coiL The mean radius of the coil is one n^ of the radius of a circle, whose circumference is the sum of the circumferences of all the actual circles formed by the wire ; and if the circles are evenly distributed, so that there are the same number of turns in each layer, we can find the mean radius by taking the mean between the radius of the groove in which the wire is wound and the external radius of the last layer. Let this mean radius be r ; and suppose, moreover, that the dimensions of the groove are so small that we can neglect the squares of the ratios of the depth or breadth of the groove to the mean radius r, then it cao be shewn ^ that the magnetic force, due to a current / in the actual coil, is n times that due to the same current in a single circular wire of radius r, so that it is equal to 2nir ifr. » Maxwell, Electricity and Magnetism^ voL iL § 711. Ch, XIX.] Experiments on Electric Currents. 503 And if the magnet be also small compared with r, and the plane of the coils coincide with the meridian, the re- lation between the current 1 and the deflection ^ is given by / = H r tan ^ / (2 « »). Unless, however, the breadth and depth of the coil be small compared with its radius, there is no such simple connection as the above between the dimensions of the coil and the strength of the magnetic field produced at its centre. The strength of field can be calculated from the dimen- sions, but the calculation is complicated, and the measure- ments on which it depends are difficult to make with accuracy. Definition of Galvanometer Constant. — The strength at the centre of a coil of the magnetic field pro- duced by a unit current flowing in it, is called the galvano- meter constant of the coiL Hence, if a current 1 be flowing in a coil of which the galvanometer constant is g, the strength of the field at the centre of the coil is g i^ and the lines of force are at right angles to the coiL Let us suppose that a coil, of which the galvanometer constant is g, is placed in the magnetic meridian, with a magnet at its centre, and that the dimensions of the magnet are so small that, throughout the space it occupies, we may treat the magnetic field as uniform ; then, if the magnet be deflected from the magnetic meridian, through an angle ^ by a current iy the moment of the force on it due to the coil is g I M cos 4>y M being the magnetic moment of the magnet, while the moment of the force, due to the earth,is h m sin ^; and since these must be equal, the magnet being in equili- brium, we have /= H tan <^/a In using a tangent galvanometer it is not necessary that the earth's directing force alone should be that which retains the magnet in its position of ^qilibrium when no 504 Practical Physics. [Ch. XIX. current passes round the coil. All that is necessary is that the field of force in which the magnet hangs should be uniform, and that the lines of force should be parallel to the coils. This may be approximately realised by a suitable distribution of permanent magnets. If the coil of wire can be turned round a vertical axis through its centre, parallel to the plane of the circles, the instrument can be used as a sine galvanometer. For thb purpose place the coils so that the axis of the magnet lies in their plane before the current is allowed to pass. When the current is flowing, turn the coils in the same direction as the magnet has been turned until the axis of the magnet again comes into the plane of the coils, and observe the angle + through which they have been turned. Then we can shew, as in chap. xvilL, that / = H sin ^/a To obtain these formulas, we have supposed that the dimensions of the magnet are small compared with those of the coil. If this be not the case, the moment of the force produced by the magnetic action of the coil when used as a tangent galvanometer is not mgcos^ as above, but in- volves other terms depending on the dimensions o( and distribution of magnetism in, the magnet In order to measure the deflexions, two methods are commonly in use. In the first arrangement there is attached to the magnet, which is very small, a long pointer of glass, aluminium, or some other light material This pointer is rigidly connected with the magnet, either parallel to or at right angles to its axis, and the two, the magnet and pointer, turn on a sharp-pointed pivot, being supported by it at their centre, or are suspended by a fine fibre free fix>m torsion. A circle, with its rim divided to degrees, or in good instruments to fractions of a degree, is fixed in a horizontal plane so that the axis of rotation of the magnet passes through its centre, and the position of the Ch. XIX.] Experiments on Electric Currents. 505 magnet is determined by reading the division of this circle with which the end of the pointer coincides. In some cases the end of the pointer moves just above the scale, in others the pointer is in the same plane as the scale, the central portion of the disc on which the graduations are marked being cut away to leave space for it, and the gradua- tions carried to the extreme inner edge of the disc. With the first arrangement it is best to have a piece of flat mirror with its plane parallel to the scale, beneath the pointer, and, when reading, to place the eye so that the pointer covers its own image formed by reflexion in the mirror. The circle is usually graduated, so that when the pointer reads zero, the axis of the magnet is parallel to the plane of the coils if no current is flowing. In order to eliminate the effects of any small error in the setting, we must proceed in the following manner :— Set the galvanometer so that the pointer reads zero, pass the current through it, and let be the deflexion observed* Reverse the direction of the current so that the needle may be deflected in the other direction ; let the deflexion be ^ If the adjustments were perfect — the current remaining the same — we should have 6 and ^ equal; in any case, the mean, \{0'\-&\ will give a value for the deflection corrected for the error of setting. To obtain a correct result, however, the position of both ends of the pointer on the scale must be read. Unless the pointer is in all positions a diameter of the circle, that is, unless the axis of rotation exactly coincides with the axis of the cirde, the values of the deflexions obtained from the readings at the two ends will difier. I^ however, we read the deflexions tf, dj, say, of the two ends respectively, the mean i(tf+tfi), will give a value of the deflexion corrected for errors of centering.' Thus, to take a reading with a galvano- meter of this kind, we have to observe four values of the deflexions, viz. two, right and left of the zero respectively, ■ See Godfray's Astronomy^ § 93. 5o6 Practical Physics. [Ch. xdl for each end of the needle. This method of reading should be adopted whether the iastrument be used as a tangent or a sine galvanometer. The second method of measuring the deflexion has been explained at full length in the account of the last experiment (p. 497). A mirror b attached to the magnet, and the motions of the magnet observed by the reflexion by it of a spot of light on to the scale. The following mocUflcation of this method is sometimes useful' A scale is fixed fi^cing the mirror, (which should in this case be plane) and parallel to it. A virtual image of this scale is formed by reflexion in the mirror, and this image is viewed by a telescope which is pointed towards the mirror from above or below the scale. The telescope has cross-vmres, and the measurements are made by observing the division of the scale, which appears to coincide with the vertical cross-wire, first without, and then with a current flowing in the coiL For details of the method of observation see § 23. In the best tangent galvanometers ' there are two coils, of the same size and containing the same number of turns, placed with their planes parallel and their centres on the same axis. The distance between the centres of the coils is equal to the radius of either, and the magnet is placed with its centre on the axis midway between the two coils. It has been shewn * that with this arrangement the fleld of force near the point at which the magnet hangs is more nearly uniform than at the centre of a single coil It has al^ been proved that in this case, if o be the galvanometer con- stant, n the number of tiuns in the two coils, r the mean radius, and i the depth of the groove fiUed by the wire, then " Sec § 23, p. 191. * Helmholtz's arrangement, Maxwell, EUctruity and Magnetism, roL iu § 715. • Maxwell, ElectruUy and Magnetism^ vol ii. { 713. Ch. XIX. J Experiments on Ekctric Currents. 507 Various other forms of galvanometers have been devised for special purposes. Among them we may refer to those which are adapted to the measurement of the large currents required for the electric light An accoimt of Lord Kelvin's galvanometers arranged for this purpose will be found in Professor Gray's book on ' Absolute Measurements in Elec- tricity and Magnetism ; ' while a paper by Professor Ayrton and Dr. Simpson, ' Phil Mag./ July 1890, contains valuable information about other instruments. On the Reduction Factor of a ■ Galvanometer. The deflexion produced in a galvanometer needle by a given current depends on the ratio h/g, h being the strength of the field in which the needle hangs when undisturbed, and G the strength of the field due to a unit current in the coiL This ratio is known as the reduction factor of the galvanometer. Let us denote it by ^, then k = h/o ; and if the instrument be used as a tangent galvanometer we have i = ^tan^; if it be used as a sine galvanometer / =B ^ sin 1^1 ^ and ^ being the deflexions produced in either case by a current /. It must be remembered that the reduction factor depends on the strength of the magnetic field in which the magnet hangs as well as on the galvanometer constant There is generally attached to a. reflecting galvanometer a controlling magnet capable of adjustment The value of k will accordingly depend on the position of this control magnet, which in most instruments is a bar, arranged to slide up and down a vertical axis above the centre of the coils, as well as to rotate about that axis. The sensitiveness 5o8 Practical Physics. [Ch. XDL of the instrument can be varied by varying the positicm of this magnet On the Sensitiveness of a Galvanopteter, The sensitiveness of a galvanometer will depend on the couple which tends to bring the needle back to its position of equilibrium, and is increased by making that couple smalL The couple is proportional to the magnetic moment of the needle and to the strength of the field in which the magnet hangs. Two methods are employed to diminish its value. If the first method be adopted two needles are em- ployed. They are mounted, parallel to each other, a shoct dis^nce apart, so that they can rotate together as a rigid system about their common axis. Their north poles are in opposite directions, and their magnetic moments are made to be as nearly equal as possible. If the magnetic mommts of the two be exactly the same, and the magnetic axes in exactly opposite directions, such a combination when placed in a uniform magnetic field will have no tendency to take up a definite position. In practice this condition of absolute equality is hardly ever realised, and the combination, if free to move, will be urged to a position of equilibrium by a force which will be very sipall compared with that which would compel either magnet separately to point north and south. It will take, therefore, a smaller force to disturb the com- bination from that position than would be required for either magnet singly. Such a combination is said to be astatic When used for a galvanometer the coils are made to surround one needle only; the other is placed outside them, either above or below as the case may be. The magnetic action of the current affects mainly the enclosed magnet ; the force on this is the same as if the other magnet were not present, and hence, since the con- trolling force is much less, the deflexion produced by a given current is much greater. This deflexion b still furth» Ch. XIX.] Experiments on Electric Currents, 509 increased by the slight magnetic action between the current and the second magnet In some cases this second magnet is also surrounded by a coil, in which the current b made to flow in a direction opposite to that in the first coil, and the deflexion is thereby stiU further augmented. In the second method the strength of the field in which the needle hangs is reduced by the help of other magnets ; if this method be adopted, the advantages of an astatic combination may be partly realised with an ordinary galvanometer by the use of control magnets placed so as to produce a field of force opposite and nearly equal to that of the earth at the point where the galvanometer needle hangs. The magnetic force tending to bring the needle back to its equilibrium position can thus be made as small as we please — neglecting for the moment the efiect of the torsion of the fibre which carries the mirror — ^and the de- flexion produced by a given current will be correspondingly increased. The increase in sensitiveness is most easily determined, as in § 69, by observations of the time of swing, for if h represent the strength of the field in which the magnet hangs, we have seen (§ 69) that Ha=4 w* k/m t*, m being the magnetic moment, k the moment of inertia, and t the time of a complete period But, being small, the deflexion pro- duced by a given current, on which, of course, the sensitive- ness depends, is inversely proportional to h ; that is, it is directly proportional to the square of t. The method of securing sensitiveness thus by the use of a control magnet is open to the objection that the small variations in the direction and intensity of the earth's mag- netic force, which are continually occurring, become very appreciable when compared with the whole strength of the field in which the magnet hangs. The sensitiveness, and, at the same time, the equilibrium position of the magnet, are, therefore, continually changing. 5IO Practical Physics. (Ch. XDC On the Adjustment of a Reflecting Galvanometer. In adjusting a reflecting galvanometer, we have first to place it so that the magnet and mirror may swing quite freely. This can be attained by the adjustment of the leveUing screws on which the instrument rests. There is generally a small aperture left in the centre of the coils opposite to that through which the light is admitted to the mirror. This is closed by a short cylinder of brass or copper which can be withdrawn, and by looking in from behind, it is easy to see if the mirror hangs in the centre of the coils as it should do. The lamp and scale are now placed in front of the mirror, the plane of the scale being approximately parallel to the coils, and the slit through which the light comes rather below the level of the mirror. The magnet and mirror are adjusted, by the aid of the control magnet, until the light is reflected towards the scale. The position of the reflected beam can easily be found by holding a sheet of paper close to the mirror so as to receive it, moving the paper about without intercepting the incident beam. By moving the con^l magnet, and raising or lowering the scale as may be required, the spot may be made to fall on the scale. The distance between the galvanometer and scale must now be varied until the image formed on the scale is as clear and distinct as possible ; and, finally, the control magnet must be adjusted to bring the spot to the central part of the scale, and to give the required degree of sensi- tiveness. As we have seen, the sensitiveness will largely depend on the position of the control magnet Its magnetic mo- ment should be such that when it is at the top of the bar which supports it, as far, that is, as is possible from the needle, the field which it alone would produce at the needle should be rather weaker than that due to the earth If this Ch. XIX. § 72.] Experiments on Electric Currents. 511 be the case, and the magnet be so directed that its field is opposite to that of the earth, the sensitiveness is increased at first by bringing the control magnet down nearer to the coils, becoming infinite for the position in which the effect of the control magnet just balances that of the earth, and then as the control magnet is still further lowered the sensitiveness is gradually decreased. The deflexion observed when a reflecting galvanometer is being used is in most cases small, so that the value of ^ measured in circular measure will be a small fraction ; and if this fraction be so small that we may neglect ^', we may put sin <^ = <^ = tan <^ (see p. 45) and we get 1 = ^<^ With a sensitive galvanometer in which the coils are close to the magnet the ratio of the length of the magnet to the diameter of the coil is considerable, and the galvano- meter constant is a function of the deflexion ; so that k is not constant for all deflexions in such an instrument, but depends on the angle ^ If, however, the deflexions em- ployed be small we may without serious error use the formula i=k<f^ and regard ^ as a constant. 72. Determination of the Bednotion Factor of a Galvanometer. If the dimensions and number of tiuns of the galvano- meter and the value of h can be measured accurately the reduction factor can be calculated We shall suppose, however, that these data cannot be directly measured, and turn to another property of an electric current for a means of determining the reduction factor. Let / be a current which produces a deflexion ^ in a galvanometer of which the reduction factor is k ; then if it be used as a tangent instrument we have i = ^tan^ and therefore, k s i/tan ^. 512 Practical Physics. [Ch. XIX^ § ya. If we can find by some other means the value oi t^irt can determine k by observing the deflexion ^ which it produces. Now it has been found that when an electric current is allowed to pass through certain chemical compounds whidi are known as electrolytes, the passage of the current is accompanied by chemical decomposition. The process is called Electrolysis ; the substance is resolved into two com- ponents called Ions\ these collect at the points at which the current enters and leaves the electrolytes respectively. The conductors by which the current enters or leaves the electrolyte are known as the Electrodes^ \ that at which the current enters the electrolyte is called the Anade^ and the component which appears there is the ^^fii^if. The conductor by which the current leaves the electrol3rte is the Kathode, and the ion which b found there is the Kathion. An appa- ratus arranged for collecting and measuring the products of electrolytic decomposition is called a Voltcuneter. Moreover, it has been shewn by Faraday (' Exp. Res.' sen vii.) that the quantities of the ions deposited either at the kathode or the anode are proportional to the quantity of electricity which has passed. If this quantity be varied the quantity of the ions deposited varies in the same ratia This is known as Faraday's law of electrolysis. Definition of Electro-Chemical Ek^invAUSNT. — The electro-chemical equivalent of a substance is the number of grammes of the substance deposited by the pas- sage of a imit quantity of electricity through an electrolyte in which the substance occurs as an ion. Thus, if in a time / a current /deposits m grammes of a substance whose electro-chemical equivalent is y, it follows from the above definition, in conjunction with Faraday's law, that m = y//, and hence i = mjy /. ' The term * electrode ' was originally applied by Famday in the •ciwe in which it is here used. Its application has now been extended, and It is employed in reference to any conductor by which electridtr enters or leaves an electrical apparatus of any sort Ch. XIX. § 72.] Experitntnts on Electric Currents, 513 If, then, we observe the amount of a substance, of known electro-chemical equivalent, deposited in time /, we can find the current, provided it has remained constant throughout the time /. If a current be allowed to pass between two plates of copper immersed in a solution of sulphate of copper, the sulphate is electrolysed and copper deposited on the kathode. The acid set free by the electrolysis appears at the anode, and combines with the copper. The quantity of copper deposited on the kathode in one second by a unit current has been found to be '00328 gramme. This is the electro-chemical equivalent of copper. The loss of weight of the anode is for various reasons found to be somewhat in excess of this. We proceed to describe how to use this experimental result to determine the reduction factor of a galvanometer. Two copper plates ^ are suspended in a beaker containing a solution of copper sulphate, by wires passing through a piece of dry wood or other insulating material which forms a covering to the beaker. The plates should be well cleaned before immersion by washing them with nitric acid, and then rinsing them with water, or by rubbing them with emery cloth, and then rinsing them with water. They must then be thoroughly dried. One of the plates must be care- fully weighed to a milligramme. On being put into the solution this plate is connected to the negative pole — the zinc — of a constant battery, preferably a DanielFs cell, by means of copper wire ; the other plate is connected with one electrode of the galvanometer. The positive pole of the battery is connected through a key with the other pole of the galvanometer, so that on making contact with the key the current flows from the copper of the battery round the galvanometer, through the electrolytic cell, depositing copper on the weighed plate, and finally passes to the zinc or negative pole of the battery. Since the galvanometer ' For details as to precautions see Gray, Absolute Measurement^ in Electricity and Magnetism^ p. 169. LL 5U Practical Physics. [Ch. xix. § 72. reading is most accurate when the deflexion is 45° (see p. 47), the battery should if possible be chosen so as to give about that deflexion. For this purpose a preliminary experiment may be necessary. It is also better if possible to attach the copper of the battery and the anode of the cell to two di the binding screws of a commutator, the other two being in connection with the galvanometer. By this means the current can easily be reversed in the galvanometer without altering the direction in which it flows in the cell, and thus readings of the deflexion on either side of the zero can be taken. The connections are shewn in fig. 58. b is the bat- tery, the current leaves the voltameter* v by the screw u, Pig. 58. entering it at the binding screw n from the commutator c This consists of four mercury cups, /, ^ , r, j, with two P -shaped pieces of copper as connectors, lip and x, ^ and r respectively be joined, the current circulates in one direc- tion round the gdvanometer ; by joining p and q^ r and i, the direction in the galvanometer is reversed. The cup r is connected with the positive pole of the battery b. Now make contact, and allow the cturent to flow through the circuit for fifteen minutes, observing the value of the deflexion at the end of each minute. If there be a commutator in the circuit as m the figure, adjust it so that ' See next page. Ch. XIX. § 72. J Experiments an Ekctric Currents. 515 the current flows in opposite directions during the two halves of the interval Let ^ be the mean of the deflexions observed. If the battery has been quite constant the de- flexions observed will not have varied from minute to minute ; in any case the deflexion must not have changed much during the interval. If any great variation shews itself, owing to changes in the battery or voltameter, the ex- periment must be commenced afresh. At the end of the fifteen minutes the weighed plate must be taken out of the solution, washed carefully, first under the tap, and then by pouring distilled water on it, and finally dried by being held in a current of hot dry air. It is then weighed carefully as before. It will be found to have increased in weight; let the increase be m grammes. Then the increase per second is m/(i5 x 60), and since the electro-chemical equivalent of copper is '00329, the average value of the current in C.G.S. units (electro-magnetic mea- sure) is »i/(6o X 15 X •00329). But if ^1 ^s ... <^i 5 be the readings of the deflexion, this average value of the current is also Tiy^(tan<^i-|-tan<^,+ .... tan^^i^). And if ^1 ^s, &C., are not greatly different, this expression is very nearly equal to itan ^ where ^ is the average value of ^1, • • . <^i5. We thus find 60 X 15 X 00329 X tan <^ If the factor is so small that the copper deposited in fifteen minutes— w grammes — is too little to be determined accurately, the experiment must be continued in the same way for a longer period. It must be remembered that the mass m is to be expressed in grammes. Instead of using a glass beaker to hold the sulphate, it is sometimes convenient to make the containing vessel L L 2 5x6 Practical Physics, [Ch. XIX, $ 7a itself one of the electrodes. Thus a copper crucible may be used as cathode, like the platinum one in PoggendorflTs voltameter ; in this the sulphate is placed, and the anode may be a rod of copper which hangs down into it. This form is shewn in the figure. We have already said that if the dimensions of the galva- nometer coil, and the number of turns of the wire of which it is composed can be determined, the value of k can be calculated, provided that the value of h be known ; or, on die other hand, H can be found from a knowledge of the dimen- sions, and of the value of k determined by experiment For if G be the galvanometer constant, r the mean radius, and n the number of turns, we have o = 2 ir «/r. Also k = h/g. Whence h = Gi = 2ir« k\r. The current, which is determined by the observations given above, is measured in C.G.S. units. The value of k gives the current which deflects the needle 45**, measured also in the same units. To obtain the value in amperes we must multiply the result by 10, since the C.G.S. unit of current contains 10 amperes. [Note. — We have supposed a copper voltameter to be used. A silver voltameter is more accurate. Directions for its use are given in § X, p. 579.] Experiment, — Determine the reduction factor of the givco galvanometer by electrolysis, comparing your result with that given by calculation. Enter the results thus — Battery 3 Daniells Gain of kathode .... '2814 gm. Deflexion, greatest , . . .46° >i least . . . 45*^30' „ mean of 15 . . . 45^*50' Time during which experiment lasted 15 minutes Value of ^ •0923C.G.S.uiii Radius of wire • • . . , i6*2 cm. Ch. XIX. §71.) Expertments en Electric Currents. 517 Number of turns . ... 5 V&lue of H -180 Value of >( calculated . . . -0928 f3. Faraday's Lav. Comparison of IHectro-CbeiDioal Eqaivalents. The electro-chemical equivalent of an element or radicle in absolute vieasvrt is the number of units of mass of the element or radicle separated from one of its compounds by the passage of an absolute unit of electricity. The ratio of the electro-chemical equivalents of two elements may thus be found by determining the mass o* each element deposited by the same quantify of tlectricity. In order to ensure that the same quantity of electricity passes through two solutions we have only to include both in one circuit with a battery. This plan is to be adopted in the following experiment to compare the electro-chemical equivalents of hydrogen and copper. Arrange in circuit with a battery (fig. 59) (the number of ceils of which must be estimated from the resistance' to be overcome, and must ric s» be adjusted so as to give a supply of bub- bles in the water vol- tameter that will form a measurable amount of gas in one hour) ( j) a beaker u of cop- per sulphate, in which dip two plates of cop- per c, c', soldered to copper wires passing through a piece of wood which acts as a support on lop of the beaker, and (a) a water voltameter' v. a euilf put toeether ii shewn In the 5 1 8 Practical Physics. (Ch. xix, § n Mount over the platinum plate p', by which the current is to leave the voltameter, a burette to be used for measuring the amount of hydrogen generated during the experiment, taking care that all the hydrogen must pass into the burette: Place a key in the circuit, so that the battery may be throvn in or out of circuit at will. The zinc of the battery must be in connection with the plate c' on which copper is to be deposited. The copper or platinum is in connection with the platinum plate p, od which oxygen will be deposited About three storage cells will probably be required for a supply of gas that can be measured in a convenient time ; and as this will correspond to a comparatively large curreDt, the plates of copper should be large, say 6 in. x 3 in., or the deposit of copper will be flocculent and fall off the plate. When the battery has been properly adjusted to give a current of the right magnitude, the apparatus will be in a condition for commencing the measurements. Accordingly, take out, dry, and carefully weigh the copper plate 00 which the metal will be deposited during the experiment This of course is the plate which is connected with die negative pole of the battery. Let its weight be w. After weighing the copper plate no current must be sent through the voltameter containing it, except that one which is to give the required measurement Read the position of the water in the burette — the height in centimetres of the water in the burette above the level of the water in the voltameter. Let this be k Read the barometer; let the height be h. Read also a thermometer in the voltameter ; let the temperature be i"C Make the battery circuit by closing the key and allow the figure. The plate V is inside a porous pot, such as is used ta 1 Leclanch^ battery, and the open end of the burette is sealed into the top of the pot by means o^ pitch or some kind of insulating cenxsL The hydrogen is formed inside the pot and rises into the burette. A graduated Hofmann voltameter is of course better, but the above cu be made in any laboratory with materials which are always at hand. Cm. XIX. § 73.] Experiments on Electric Currents. 519 current to pass until about twenty centimetres of the burette have been filled by the rising gas. Shut off the current, and dry and weigh the same plate of copper again ; let the weight be w'. Then the amount of copper deposited by the current is w'— w. Read again the position of the water in the burette. From the difference between this and the previous reading we may obtain the volume of the gas generated. Let the difference in volume actually observed be v cubic centi- metres, and let the height of the water in the burette above that in the voltameter at the end of the experiment be^'. Before using v to find the mass of hydrogen deposited we have to apply several corrections. There was some gas above the water in the burette before the experiment began. The pressure of the gas above the water has been increased by the experiment, and this gas has in consequence decreased in volume. We require to find what the decrease is. Let the original volume of the gas be r. The gradua- tions on the burette are generally not carried to the end, and to find v we require to know the volume between the last graduation and the tap of the burette. For this purpose a second burette is needed. This is filled with water to a known height. The burette to be used in the experiment is taken and inverted, being empty. Water is run into it from the second burette until it is filled up to the first graduation ; the quantity of water so run in is found by observing how far the level in the second burette has fallen. Or, if it be more convenient, the method may be reversed; the second burette being partly filled as before, the first burette is also filled up to some known graduation, and all the water which it contains is run out into the second ; the rise in level in this gives the quantity of water which has run out, and from this we can find the volume required i 2b Practical Pltysics. [Ch. XIX. \ 73. between the bottom of the burette and the first graduation ,* knowing this we find the vohime v easily. Now this gas of volume v was at the commencement under a pressure equal to the difference between the atmospheric pressure and the pressure due to a column of water of height /^ ; if S be the specific gravity of mercury, the pressure due to a column of water of height A is the same as that due to a column of mercury of height A/8; so that H being the height of the barometer, the pressure of the gas will be measured by the weight of a column erf mercury of height h— A/S, while at the end of the experi- ment the pressure is that due to a column h — ^'/S. Therefore the volume which the gas now occupies is h so that the decrease required is h8-^\ A-A' / h8-A\ A-A' and A' being small compared with hS, we may write this:— A -A' h8 This must be added in the observed volume v to find the volume occupied by the gas electrolysed, at a pressure due to a column of mercury of height h— A'/S, giving us thus as the volume, v + 7^ — R-. h8 It is sometimes more convenient to avoid the necessity for this correction by filling the burette with water before beginning, so that t/, the space, initially filled with gas is tJH. XIX. §73.] Experiments an Electric Currents. 521 zero. If this plan be adopted we shall still require to know the volume between the end of the burette and the gradua- tions, and this must be obtained as described above. Correction for aqueous vapour, — The solution of sul- phuric acid used in the voltameter is exceedingly dilute, and it may be supposed without error that the hydrogen gas comes off saturated with aqueous vapour ; the pressure o\ this vapour can be found from the table (34), for the tem- perature of the observation, f C Let it be e. Then if ^ be expressed as due to a column of mercury of e centimetres in height, the pressure of the hydrogen will be measured by and its volume at this pressure and temperature / is v+r h8 Thus its volume at a pressure due to 76 centimetres and temperature o** C is I^t this be v'. The weight required is v' x '0000896 gm., •0000896 being the density of hydrogen. But according to Faraday's fundamental law of electro- lysis, the weights of two elements deposited by the same current in the same time are proportional to their chemical equivalents. We must, therefore, have w -"■ w — ;?— r= chemical equivalent of copper. v' X '0000896 The value of the equivalent, as deduced from chemical experiments, is 3175. Experiment, — Determine by the use of voltameters the chemical equivalent of copper. 522 Practical Physics. [Ch. XIX. 9 73. Enter results thus : w ■61*0760 gms. ^-20 cm. \ w'-6i'i246gms. A'- 5 cm. V -18-5 cc # - 1-9 cm. V « 1-25 cc / -15** C H « 75*95 cm. v'-i7-occ Chemical equivalent -31*13 74. Joule's Law—Measnrement of Eleotromotiye Foree. We have seen that work is done when a quantity of electricity passes from a point at one potential to a second point at a different one. If q be the quantity of electricity which passes thus, and s the difference of potential, or electromotive force, maintained constant between the points while Q passes, then the work done is Q x b. If the electricity pass as a steady current of strength c, for a time / seconds, then, since the strength of a current is measured by the quantity which flows in a unit of time, we have Q = cii and if w be the work done, W = EC/. If this current flow in a wire the wire becomes heated, and the amount of heat produced measures the work done, for the work which the electricity does in passing from the point at high to that at low potential is transformed into heat If H be the amount of heat produced and j the mechanical equivalent of heat, that is, the number of units of work which are equivalent to one unit of heat, then the work required to produce h units of heat is j h. Hence we have JH=W = EC/; whence E = JH/ (c/). Now J is a known constant, H can be measured by immersing the wire in a calorimeter (see § 39) and noting the rise of temperature of a weighed quantity d \ Ch. XIX. $ 74.] Experiments an Electric Currents, 523 water which is contained therein ; if a copper-voltameter be included in the circuit c / is obtained, knowing the electro-chemical equivalent of copper, by determining the increase in weight of the cathode. We can thus find b, the difference of potential between the two points at which the current respectively enters and leaves the wire in the calorimeter. For the calorimeter we use a small vessel of thin sheet copper polished on the outside and suspended in another copper vessel, as in § 39. The water equivalent of this must be determined, as is explained in that section, either experimentally or by calculation from the weight of the vessel and the known specific heat of copper, which for this purpose may be taken as *i. A small stirrer made of thin copper wire coiled into a spiral may be included in the estimate with the calorimeter determination. The outer vessel of the calorimeter is closed by a copper lid with a hole in the middle, through which a cork passes. The end of the stirrer passes through a hole in this cork, and through two other holes pass two stout copper wires, to the ends of which the wire to be experimented on is soldered The thermometer is inserted through a fourth hole. The bulb of the thermometer should be small, and the stem should be divided to read to tenths of a degree. The wire should be of German-silver covered with silk and coiled into a spiral. Its length and thickness will depend on the nature of the source of electromotive force used If we take a battery of three storage cells then for con- venient working, it will be found that the electrical resistance of the wire (see chap, xx.) should be about 4 ohms. The two ends are soldered on to the copper electrodes and the wire completely immersed in the water of the calorimeter. It must be carefully remembered that the quantity which we are to determine is the difference of potential between the two points at which the wire cuts the surface of the water. Some of the heat developed in the wire will of course remain in it, and in our calculations we ought strictly to 524 Practical Physics. [Ch. xix. § 74. allow for this. It will be found, however, that in most instances the correction is extremely small, and may, for the purposes of the present experiment, be safely neglected. We may assume that the whole of the heat produced goes into the water and the calorimeter. But the experiment lasts for some time, and meanwhile the temperature of the calorimeter is raised above that of the surrounding space, so that heat is lost by radiation. We shall shew how to take the observations so as to compensate for this. The apparatus is arranged as follows (fig. 60) : — The cathode c of the vol- ^^^ ^ tameter v is carefully weighed and con- nected to the nega- tive pole of the bat- tery B, the anode c' being connected by means of a piece of copper wire with one of the ends erf the wire in the calorimeter a ; the other end of thb wire is joined through a key k to the positive pole of the battery. The plates of the voltameter must be so large and so close together that its resistance maybe very small indeed compared with that of the wire in the calorimeter : otherwise the rise of temperature in the calorimeter may be hardly large enough for convenient measurement without using a considerable number of battery cells. To perform the experiment, note the temperature of the water and allow the current to flow, keeping the water well stirred ; the temperature will gradually rise. After two minutes stop the current; the temperature may still rise slighdy, but if the stirring has been kept up, the rise, after the current has ceased flowing, will be very small. I^t the Ch. XIX. 5 74.] Experttnents on Electric Currents, 525 total rise observed be rj degrees. Keep the circuit broken for two minutes; the temperature will probably fall. Let the fall be t^ degrees. This fall during the second two minutes is due to loss of heat by radiation ; and since during the first two minutes the temperature did not differ greatly <rom that during the second two, we may suppose that the loss during the first two minutes was approximately the same as that during the second two ; so that, but for this loss, the rise of temperature during those first two minutes would have been ti +T2 degrees. We thus find the total rise of temperature produced in the mass of water in two minutes by the given current by adding together the rise of temperature during the first two minutes and the fall during the second two minutes. Take six observations of this kind, and let the total rise of tempera- ture calculated in the manner above described be t degrees ; let the mass of water, allowing for the water equivalent of the calorimeter and stirrer, be m grammes, then the quantity of heat given out by the current in twelve minutes is /« t units. The experiment may also be performed by allowing the current to run continuously and determining the radiation loss as on p. 292. Let M grammes of copper be deposited by the same current ; then since the passage of a unit of electricity causes the deposition of '00328 gramme of copper, the total quantity of electricity which has been transferred is m/'oo328 units, and this is equal to 0/ in the equation for e. Hence E = J »i T X •00328/M. Now the value of j in C.G.S. units is 42 x lo*, so that we have E = 420X328X»«X t/m. The value of e thus obtained will be given in C G.S. units ; the practical unit of KM.F. is called a volt, and one volt con- tains lo* CG.S. units ; hence the value of e in volts is 420 X 328 X m X t/(m X 10*). 526 Practical Physics. [Ch. XIX. § 74- We have used the results of the experiment to find e. If, however, e can be found by other means — and we shall see shortly how this may be done — the original equa- tion, JH=EC/, may be used to find j or c It was first employed by Joule for the former of the two purposes, Le. to calculate the mechanical equivalent of heat, and the law expressed by the equation is known as Joule's law. Experiment — Determine the diflTerence of potential between the two ends of the given wire through which a current is flowing. Enter results thus : — Mass of water . . • • • 24*2 gms. Water equivalent of the calorimeter . 4*2 gms. m • . • • 28*4 gms. M . . . . '222 gm. Total nse of temperature for each two minutes :— j"'-4 4'-4 4"'-3 4" • • . • • 30-8 24^-8 E - 4-37 X lo' - 4-37 volts. CHAPTER XX. ohm's law — COMPARISON OF ELECTRICAL RESISTANCES AND ELECTROMOTIVE FORCES. We have seen that if two points on a conductor be at different potentials, a current of electricity flows through the conductor. As yet we have said nothing about the relation between the difference of potential and the current produced. This is expressed by Ohm's law, which states that the current flowing between any two points of a conductor is direcdy proportional to the difference of potential between those points so long as the conductor joining them remains the same and in the same physical state. Thus, if c be the current, iand e the electromotive force, c is proportional to B, and we may write n ^^ R Ch. XX.J Ohm's Law. 527 where r is a quantity which is known as the resistance of the conductor. It depends solely on the shape and tempera- ture of the conductor, and the natiure of the material of which it is composed, being constant so long as these re- main unaltered. Definition of Electrical Resistance. — It is found by experiment that the ratio of .the E.M.F. between two points to the current it produces, depends only on the con- ductor which connects the two points ; this ratio is called the resistance of the conductor. The reciprocal of the resistance — that is, the ratio of the current to the electromotive force — is called the conductivity of the conductor. Thus between any two points on a conductor there is a certain definite resistance : a metal wire, for example, has an electrical resistance of so many units depending on its length, cross-section, material, and temperature. Resistance coils are made of such pieces of wire, covered with an insulating material, cut so as to have a resistance of a - certain definite number of units and wound on a bobbia The ends of the coil are fastened in some cases to bind- ing screws, in others to stout pieces of copper which, when the coil is in use, are made to dip into mercury cups, through which connection is made with the rest of the apparatus used. We refer to § 78 for a description of the method of employing such coils in electrical measurements. Standards of resistance have the advantages of material standards in general. The resistance is a definite property of a piece of metal, just as its mass is. The coil can be moved about from place to place without altering its resistance, and so from mere convenience electrical resist- ance has come to be looked upon as in some way the fundamental quantity in connection with current electricity. We have defined it by means of Ohm's law as the ratio of electromotive force to the current Whenever difference of potential exists between two points of a conductor, a current 528 Practical Physics, £Cr. XX. of electricity is set up, and the amount of that current de- pends on the E.M.F. and the resistance between the points. We may say that electrical resistance is that property of a conductor which prevents a finite electromotive force fitjm doing more than a finite quantity of work in a finite time. Were it not for the resistance, the potential would be instan- taneously equalised throughout the conductor; a finite quantity of electricity would be transferred fi-om the one point to the other, and therefore a finite quantity of work would be done instantaneously. The work actually done in time / is, we have seen, equal to E c/, and by means of the equation c = e/r expressing Ohm's law, we may write this w = CE / = E* //r = c* r /. Moreover the E.M.F. between two points is given if we know the resistance between them and the current, for wc have E = c R. Further the resistance of a wire is evidently equal to the rate of expenditure of energy required to in.iin- tain unit current in the wire. On the Resistance of Conductors in Series and Multiple Arc, If A B, B c be two conductors of resistances r, and r,, the resistance between a and c is R| + r,. For let the potentials at a, b, c be Vj, V3, V3 respectively, and suppose that owing to the difference of potential a current / is flowing through the conductors. This current is the same in the two conductors (see p. 492), and if r be the resist- ance between a and c, we have from Ohm's law v,-Va = R,/ Vj-V3 = R,/ Vi-V3=R /. But by adding the first two equations we have Vi-V3 = (Ri +R2)/; /. R =s Ri + Rf Ch. XX.] Ohm's Law. 529 By similar reasoning it may be shewn that the resultant resistance of any number of conductors placed end to end is equal to the sum of the resistances of the several con- ductors. Conductors connected in this manner are said to be in series. Again, let there be two conductors of resistances Ri, R^i joining the same two points a and b, and let r be the equivalent resistance of the two, that is, the resistance of a conductor, which, with the same E.M.F. would allow the passage of a current of electricity equal to the sum of those which actually flow in the two conductors. Hence, if v„ v, be the potentials at a and b, we have Vi-V2__^. , v,-v,_. . and R 1 ^" "r. V| R = / = /, 4/» • R R, R, R = RlRj R, + R, = Rj/Ri- Also Conductors joined up in the above manner are said to be connected in multiple arc ; thus, remembering that the reciprocal of the resistance is called the conductivity, we may shew by reasoning precisely similar to that given above that the conductivity of a system .of any number of con- ductors m multiple arc is the sum of the conductivities of the several conductors. Let B A c be a circuit including a battery b, and suppose that we wish to send between the two points, a and (^ only i/nih part of the current produced by the battery. Let R be the resistance between a and a Connect these two points by a second conductor of resistance, r/(«— *)• MM 530 Practical Physics. ich. XX. Let I'l be the current in the original conductor between A and c, /j the current in th^ new conductor, i ihe current in the rest of the circuit Then we have /5/f,=R(«-/)/R; and So that The second conductor, connected in this manner with the two points, is called a shunt, and the original circuit is said to be shunted. Shunts are most often used in connection with galvano- meters. Thus we might require to measure a current by the use of a tangent galvanometer, and, on attempting to make the measurement, might find that the galvanometer was too sensitive, so that the deflexion produced by the cur- rent was too large for measurement By connecting the electrodes of the galvanometer with a shunt of suitable resistance we may arrange to have any desired fraction of the current sent through the galvanometer. This fraction can be measured by the galvanometer, and the whole current is obtained from a knowledge of the resistances of the shunt and galvanometer. A galvanometer is often fitted with a set of shunts, having resistances 1/9, 1/99, and 1/999 of its own resistance, thus enabling •!, -oi, or -GO I of the whole current to be transmitted through it In applying Ohm*s law to a circuit in which there is a battery of electromotive force e, it must be remembered that the battery itself has resistance, and this must be in- cluded in the resistance of the circuit Thus, if we have a circuit including a resistance R, a battery of E.M.F. k and resistance b, and a galvanometer of resistance c, the total resistance in the circuit is r+b+g, and the current is e/(r+b4-g. Ch. XX.] Ohm*s Law. 531 The normal E.M.F. of the battery is taken to be the difference of potential between its poles when they are insulated from each other. If they be connected together, the difference of potential between them will depend on the resistance of the conductor joining them. In the case in point this is R+o ; and since the difference of potential is found by multiplying together the current and the resist- ance, it will in that case be e(r+g)/(r+g+b). On the Absolute Measurement of Electrical Resistance, Electrical resistance is measured in terms of its proper unit defined by the equation For let a conductor be such that unit difference of potential between its two ends produces unit current ; then in the above equation s and c are both unity ; so that r is also unity and the conductor in question has unit re- sistance. Definition of an Absolute Unit Resistance. — The unit of resistance is the resistance of a conductor in which unit electromotive force produces unit current This is a definition of the absolute unit Now it is found' that on the C.G.S. system of units the unit of resistance thus defined is far too small to be convenient There- fore, just as was the case for E.M.F., a practical unit of resistance is adopted, and this contains 10^ absolute CG.S. units, and is called an 'ohm'; so that i ohm contains 10^ absolute units. We have already seen that the volt or practical unit of E.M.F. is given by the equation I volt =10^ absolute units. » Sec F. Jenkin, ElectrUUy and Magnetism^ chap. x.j Maxwell. Electricity and Matrnitism, yol. ii. § 629. M M2 532 Practical Physics. £Ch. xx. Let us suppose that we have a resistance of i ohm and that an KM. F. of I volt is maintained between its ends; then we have for the current in absolute units c = - = — - = — absolute unit = i ampbre. R lo' lO ^ Thus an ampere, the practical unit of current, is that produced by a volt when working through an ohm. But electrical resistance is, as we have seen, a property of material conductors. We can, therefore, construct a coil, of German- silver or copper wire suppose, which shall have a resistance of i ohm. The first attempt to do this was made by the Electrical Standards Committee of the British Association, and the standards constructed by them are now at the National Physical Laboratory. More recent experiments have shewn, however, that these standards have a resistance somewhat less than I ohm. They have for some time past been in use as ohms and numbers of copies have been made and circu- lated among electricians. The resistances of these standards are now known as British Association Units. In accordance with the resolutions of the Committee of the British Association passed at Edinburgh in 1892, it has been decided to define the ohm in terms of the resistance of a certain column of mercury at the temperature of melt- ing ice. The length of the column is 106*3 centimetres ; for practical purposes the area of its cross-section is one square millimetre, but the area of such a column would always be determined by finding the mass of a known length, and dividing this by the density of mercury in grammes per ac and by the length. Now the specific gravity of mercury is known with all the necessary accuracy, but the density of water is still a little uncertain. To avoid the difiSculty caused by this, the mass of mercury in the column is stated The specific gravity of mercury at o*' is 13*5956 ; if we assume that the mass of one cc of water at 4"* C is one gramme, then the mass of a column of mercury 106-3 cm Ch. XX.] OknCs Law, 533 long, one square mm. in section, is 13*5956 x i'o63 grammes, and this comes to 14*452 1 grammes. Thus one ohm is defined to be the resistance of 14*4521 grammes of mercury in the form of a column of uniform cross-section 106-3 cm. in length at o® C Moreover it has been shown by experiment that I B.A. unit = '9866 ohm. Thus I ohm = 101358 B.A. unit On Resistance Boxes, For practical use resistance coils are generally grouped together in boxes. The top of the box is made of non- conducting material, and to it are attached a number of stout brass pieces shewn in fig. 61 at a, b, c, d. A small space is left be- fig 6x. tween the con- secutive brass pieces, and the ends of these pieces are ground in such a way that a taper plug of brass can be inserted between them and thus put the two consecutive pieces into electrical connection. The coils themselves are made of German-silver or platinum-silver wire. The wire is covered with silk or some other insulating material. A piece of wire of the required resistance is cut off and bent double. It is then wound on to a bobbin of ebonite or other insulating material The bobbins are not drawn in the figure. The two ends are soldered to two con- secutive brass pieces in the box, the bobbin being fixed to the under side of the lid of the box. The coils when complete are covered with paraffin to maintain a good insulation. Let A, B be the two brass pieces, and suppose a current flowing from a to b ; if the plug is in its place, the current 534 Practical Physia. [Ch. XX. can pass through it, and the resistance between a and d is infinitesimally small, provided always that the plug fits properly. If, however, the plug be removed, the current hai to flow through the coil itself ; so that by removing the plug the resistance of the coil may be inserted in the circuit between a and b. The coils in a box are generally arranged thus : — 1225 10 10 20 50 100 100 200 500 units, &c Thus, if tliere be the twelve coils as above, by taking out suitable plugs we can insert any desired integral number of units of resistance between i and 1000, like weights in the balance. Binding screws, s, s', are attached to the two extreme brass pieces, and by means of these che box can be connected with the rest of the circuit The coils are wound double, as described, to avoid the effects which would otherwise arise from self-induction,' and also to avoid direct magnetic action on the needle of the galvanometer. On the Relation between the Resistance and Dimensions of a Wire of given Material, AVe have seen that if two conductors be joined in series the resistance of the combination is the sum of the resist- ances of the parts. Let the conductor be a long wire of uniform material and cross-section. Then it follows from the above (p. 528) that the resistance is proportional to the length; for if we take two pieces of the same length they will have the same resistance, and if connected end to end the resist- ance of the double length is double that of the single. Thus the resistance is proportional to the length. Again, we may shew that the resistance is inversely proportional to the area of the cross-section. For suppose two points, A and b, are connected by a single wire, the J7/ ' ^^^^;/- Thompson's Eke. and Mag,, § 404 ; J. J. Thomson. EUc. and Mag., chap. xi. Ch. XX.] Ohnfs Law. 535 resistance of which is r. Introduce a second connecting wire of the same length and thickness, and therefore of the same resistance as the former. The resistance will now be—, and since it was found by Ohm that the resistance 2 depends on the area of the cross-section and not on its form, we may without altering the result suppose the two wires, which have been laid side by side, welded into one, having a cross-section double of that of either wire Thus, by doubling the cross-section the resistance is halved. The resistance, therefore, varies inversely as the area of the cross-section. Definition of Specific Resistance. — Consider a cube of conducting material having each edge one centimetre in length. Let two opposite faces of this be maintained at different potentials, a current will be produced through the cube, and the number of units in the resistance of the cube is called the specific resistance of the material of which th(« cube is composed. Let p be the specific resistance of the material of a piece of wire of length / and cross-section a, and let r be the resistance of the wire. Then R = p Ija, For, suppose the cross -section to be one square centimetre, then the resistance of each unit of length is p and there arc / units in series ; thus the whole resistance is p /. But the resistance is inversely proportional to the cross-section, so that if this be a square centimetres, the resistance R is given by the equation R = p Ha, Again, it is found that the resistance of a wire depends on its temperature, increasing in most cases uniformly with the temperature for small variations, so that if R© be the resistance at a temperature zero and r that at temperature t^ we have R = R^(l+a/), 536 Practical Physics, [Cii. XX where a is a constant depending on the nature of the material of the wire ; this constant is called the tempexature coefficient of the coil. For most materials the value of a is small. German- silver and platinum-silver alloy are two substances for which it is small, being about "00032 and •00028 respectively, while for special alloys such as * Eureka' and * Manganin * its value has been reduced to about one- tenth of these figures. Its value for copper is considerably greater, being about •003, and this is one reason why resistance coils are made of one of the above alloys in preference to copper. Another reason for this preference is the fact that the specific resistance of the alloys is much greater than that of copper, so that much less wire is necessary to make a coil than v^ required if the material be copper. 75. Comparison of Electrical Besifltances. Ohm's law forms the basis of the various methods em- ployed to compare the electrical resistance of a conductor with that of a standard coil. In the simplest arrangement of apparatus for making the measurements the connections are made in the following Fig. 6a manner (fig. 62) : — One pole of a battery b of constant E.M.F. is connected to one end a of the conductor whose resistance is required ; the other end c of this conductor is in connection with a resistance box M N. N is in con- Ch. XX. § 75.] Ohnis Law. 537 nection with a key or, better, a commutator K, from which the circuit is completed through a galvanometer o to the other pole of the battery. Let X be the resistance to be measured, b the battery resistance, G that of the galvanometer, and suppose a resistance R is in circuit in the box. Make contact with the commutator. A current passes through the galvanometer. Observe the deflexion when the needle has become steady. Reverse the commutator ; the galvanometer needle is deflected in the opposite direction, and if the adjustments were perfect, the two deflexions would be the same. They should not differ by more than 0^*5. Adjust R, the resistance in the box, if it be possible, until the deflexion observed is about 45**, Of course it may be impossible to do this with the means at hand. If when R is zero the deflexion observed be small, the electro- motive force of the battery will require to be increased ; we must use more cells in series. \% on the other hand, with as great a resistance in the box as is possible, the deflexion be too large, then either the galvanometer must be shunted or the E.M.F. of the battery reduced by reducing the number of cells, or by connecting its poles through a shunt. In any case the deflexion should be between 30° and 60**. Let E be the E.M.F. and k the reduction factor of the galvanometer, which, we shall suppose, is a tangent instru- ment. Then, if 1 be the current, and a the mean of the two deflexions in opposite directions, we have = / = >t tan o. B + G + X-fR Hence B-l-G-f x-f R =e/>S tan o , . . (i) and if B, G, e, and k be known, r and a being observed, this equation will give us x. If E and k be not known, while b and g are, we proceed thus. Take the unknown resistance x out of the circuit, connecting one pole of the battery with the electrode m 538 Practical Physics. [Ch. xx. § 75. of the resistance box. Take a resistance r' out of the box and observe the deflexion, which, as before, should lie between 30® and 60®, reversing the current and reading both ends of the needle ; let the mean deflexion be «'. Then we have, as before, if the battery have a constant E.M.F., . = >& tan a' : B + G-fR' /. E/^ = (B + G + R')tan a' ... (2) so that the original equation (i) becomes B+G+x+R = (B+G+R')tan a'/tano, . • • (3) and from this x can be found. But in general b and g will not be known. We can easily find the sum b+g as follows : — Make two sets of observations exactly in the same manner as the last were made, with two diflerent resistancef Ri, R2 out of the box, and let the deflexions be ay and (4 ; a J may be just over 30% aj just under 60®. [There should be a large difference between ay and a,, for we have to divide, in order to find the result, by tan aj — tan aj, and, if this be small, a large error may be produced.] Then, assuming as before that the E.M.F. of the battery does not alter, we have F = k tan a„ . . , (4) and Hence and B4-G + R| B B + G + Rj — k tan n,. . • . (5) (b + g+r,) tanai= - = (B+G + R2) tan o^ K p + n-^ Ritanai— Ratan aa ^ .^. tanaj^— tan a, • • • V / CTm. XX. § 75.] Ohm's Law. 539 Having thus found b+g, we may use either of the equa- tions ( 4) or ( 5) in combination with (i) to give us x. If we wish to find b and G separately we may proceed as follows : — Shunt the galvanometer with a shunt of resistance s ; then the resistance between the poles of the galvanometer is equivalent to gs/(s+g). Make two more observations like those from which equations (4) and (5) are deduced, we thus find a value for b+g s/(s+g). Suppose we find having already obtained b+g=^, when y is written for the right-hand side of equation (6). Hence thus or G S s + o 0» = (S + G)(y-^), G»-o(y— S) — S(>' — ^) = 0; ••. G = i|>--4rH->/ {(7-^)2 + 48 (r-ir)}]. Thus, G having been found, b is given from the equation The methods here given for measuring resistance, in- volving, as they do, the assumption that the E M.F. of the battery remains the same throughout, cannot be considered as completely satisfactory. Others will be given in §§ 77-79, which are free from the objections which may be urged against these. Various modifications of the above methods have been suggested for measuring more accurately the resistance of a battery or galvanometer. For an account of these the reader is referred to Kempe's * Handbook of S40 Practical Physics. [Ch. XX. § 75. Electrical Testing,' chapters v. and vL In practice much is gained by a little judgment in the choice of the resistances taken from the box. Thus, in finding b+g as above it mighr happen that when Rj is 19, a^ is 59** 30', and when r, is 20, aa is 58** 45'. Now the tangent of either of these angles can be looked out equally easily in the tables, but the multiplication involved in finding r^ tan a^ is much more easily done if R^ be 20 than if it be 19. Experiment. — Determine the resistance of the given co3 x. Enter results thus : — Observations to find B -f G. R, » 20 ohms. a, « $f* R,«5o „ ai-34'* Whence B + G ■ 3*37 ohms. Observations to find x. R B 10 ohms. a «-46*'53 R,-20 „ 0,-57' Whence X - 2075 ob'.ns. N.B. — If a large number of resistances have to be deter- mined by the use of the same galvanometer, it will be best to calculate the value of B + G, and the ratio of the E.M.F. to the reduction factor once for all, checking the results occasionally during the other observations. These are both given by the observations just made, for we have found B + G, and we have B — k tfln a + G-t-R, * • • « E — 1 k -(B + G + R,) tan €4. With the numbers in the above example, B + G " 3*37 ; R| « 20 ; and we find CII.XX. §75«] Ohnis Law. 541 So that, if we find, with an unknown resistance x in circuit and a resistance R out of the box, that the deflexion is a, we obtain B + G + X + R-T-1— «2i2Z, i?tan a tan a 76. Comparison of Electromotive Forces. We may moreover use Ohm's law to compare the electro- motive forces of batteries.* For suppose we have two bat- teries ; let B, b' be their resistances, e, e' their electromotive forces. Pass a current from the two batteries in turn through two large resistances, r and r' and the galvanometer, and let the deflexions observed be a, a'. Suppose the galvanometer to be a tangent instrument Then, if i be its reduction factor, G its resistance, we have E = >J(B + G+R)tan a, e' = >&(b' +G f R') tan a'. E' Ilcnce B__ (B+G+R)tantt B' (B' + G + R')tana'* and B+G, b'+g being determined as in the last section, the quantities on the right-hand side are all known. In practice there are some simpliflcations. A Thomson's reflecting galvanometer is used, and this is so sensitive that R and r' will need to be enormously large to keep the spot of light on the scale. The values will be probably from eight to ten thousand ohms if only single cells of the batteries in ordinary use be employed Now the resistance of such a cell will be very small compared with these ; an ordinary quart Daniell should be under one ohm ; a Leclanch^ from one to three ohms ; and hence we may neglect a and d' as compared with r and r', and we have B (R+G)tana E^ (r' + g) tan a^ ' See o. 531. 542 Practical Physics. [Ch. XX. | n». This equation is applied in two ways : — (i) The Equal Resistance Method. — The resistance r' is made equal to r, Le the two batteries are worked through the same external circuit, and we have then E _ tana e' "" tan a'* But if the angles a, a' be not too large, the scale-de- flexions of the spot of light are very nearly proportional to tan a and tan a'. Let these deflexions be 8 and V respec- tively, then B 8 E / Y For this method we do not need to know the galvano- meter resistance, but we suppose that the galvanometer is such that the displacement of the si>ot is proportional to the current (2) The Equal Deflexion Method. — In this method of working a! is made equal to a, and we have e' r'+g* For this method we require to know o, or, at any rate, to know that it is so small compared with r and r' that we may neglect it The method has the advantage that we do not assume any relation between the current in the galvano- meter and the deflexion produced, except that the same current produces the same deflexion ; and this is obviously true whatever be the form of the instrument Both methods are open to the objection that the E.M.F. of a battery which is actually producing a current changes from time to time. We shall see in § 80 how to compare the E.M.F. of batteries without allowing them to produce a current Experimenis, Compare the E.M.F. of the given batteries by the equal resist- ance and the equal deflexion methods, and taking the ELM.F. erf Ch, XX. § 76.] Ohm's Law, 543 the DanielPs cell as it)8 volts, find the E.M.F. of the others in volts. Enter results thus : — Equal Resistance Method, — Resistance used, 10,000 ohms. Internal resistance of cells, small. Battery Deflexions in E.M.F. b scale divisions volts Daniell ... 46 i'o8 Sawdust Daniell • . 35 *82 Leclanch^ . . . 52 1*22 Bichromate ... 68 i'6o Equal Deflexion Method. — Deflexion, 83 scale divisions. Galvanometer resistance, small. Battery Resistance E.M.F. in volts Daniell . • • 8000 ro8 Sawdust Daniell . • 6020' '81 Leclanch^ . . . 9040 r22 Bichromate. • . 11980 i'6i 77. Wheat8tone*8 Bridge. The method of comparing electrical resistances which has been already described depends on the measurement of the deflexion produced in a galvanometer, and we make the assumptions that the E.M.F. of the battery remains con- stant during the experiment, and that the relation between the current flowing through the galvanometer and the de- flexion it produces is known. The disadvantages which thus arise are avoided in the VVheatstone bridge method, the principles of which we proceed to describe. It follows from Ohm*s law (p. 526) that, if a steady cur- rent be flowing through a conductor, then the electromotive force between any two points of the conductor is propor- tional to the resistance between those points.* We can express this graphically thus. Let the straight line a b (fig. 63) represent the resistance between the two points a and B of a conductor, and let the line a d, drawn at right angles ^o A B, represent the electromotive force or diflerence of 544 Practical Physics. [Ch. XX. § 77. potential between a and & Join d b, and let m be a point on the line a b, such that a m may represent the resistance between a and another point of the conductor. Draw m l Fig. 6). at right angles to a b to meet b d in l, then l m represents the KM.F. between m and b. For if c represent the current flowing through the con- ductor, then, by Ohm's law, DA a b =?c; and since m l is parallel to d a, DA ab LM MB • • LM = CXMa But since M b represents the resistance and c the currem between two points m and b, it follows from Ohm's law that LM represents the E.M.F. between those points. Now let a' b' represent the resistance between two points on another conductor, between which the E.M.F. \s the same as that between a and b, and let a' d' represent this KM. P. ; then a' D' = A D. Join d' b', and in it take V m', such that l' m' shall be equal to LM. Then m' will represent a point on the second conductor Ch, XX. S 77.] Ohm's Law. 545 such that the difference of potential between it and b' is equal to the difference of potential between m and a Thus if B, b' be at the same potential, a, a' and m, m' re- spectively are at the same potentials. Hence, if m m' be joined through a galvanometer g, no current will flow through the galvanometer, and no deflexion, therefore, will be observed We can now express the condition for this in terms of the four resistances am, mb, a'm', m'b'. Let these resistances respectively be denoted by p, q, r, and s. Draw LN, l' n' parallel to a b and a' b'. Then clearly d n = d' n', and we have P AM N L D N d' n' N' L' a' m' R Q MB MB LM L' M' M'b' M' b' S Thus the condition required is p _R Q^s' If, then, we have four conductors, a m, m b, a' m', m' b', and we connect together b and b', and so keep them at the same potential, and also connect a and a', thus keeping them at any other common potential, then, provided the above condition holds, we may connect m and m' through a galvanometer without producing a deflexion ; and con- versely if, when m m' are thus connected, no deflexion be observed, we know that the above condition holds. Hence, if p and Q be any two known resistances, R any unknown resistance, and s an adjustable known resistance, and we vary s, the other connections being made as described, until no deflexion is observed in the galvanometer, r can be found, for we then have p R = SX -, Q and p, Q, s are known. In practice, to secure that b and b' should be at the same potential, they are connected together, and to one pole of a battery, a and a' being connected through a key, to thcother pole. 546 Practical Physics. [Ch. XX. § 77- Fig. 64 shews a diagram of the connections, ac, cb correspond to the two conductors am, m b of fig. 63, while AD, DB correspond to a' m', m' b'. a key xf is placed in the galva- nometer circuit and a A second key k in the battery circuit. On making contact with the key K a difference of potential is established between a and b, and a current flows through the two conductors a c b and a d a If on making contact with k' no deflexion is observed in the galvanometer, it follows that c and d are at the same potentiali and therefore that p R = s X -. Q In practice p, q, and s are resistance coils included in the same box, which is arranged as in fig. 65 for the pur- poses of the experiment, and is generally known as a Wheat- stone-bridge box, or sometimes as a Post-Ofl5ce box.* The ' But see next page* Ch. XX. § 77.) Ohm's Law. 547 resistances p and q, which are frequently spoken of as the arms of the bridge, are taken, each from a group of three coils of 10, 100, and 1000 units. Thus, by taking the proper plugs out we may give to the ratio p/q any of the values 100, 10, I, •!, or 'oi. The resistance s is made up of 16 coils from i to 5,000 ohms in resistance, and by taking the proper plugs out it may have any integral value between i and 10,000 units. Thus the value of r may be determined to three figures if it lie between i and 10, or to four figures if it be between 10 and 1,000,000, provided, that is, the galvanometer be sufficientiy sensitive. At A, B, c, and d are binding screws, those at a and d being double. By means of these the electrodes of the battery, galvanometer, and conductor whose resistance is required, are connected with the box. In some boxes the two keys, k and k', are permanently connected with the points A and c, being fixed on to the insulating material of the cover. The arrangement is then technically known as a Post-Office box. The galvanometer to be employed should be a sensitive reflecting instrument ; the method of adjusting this has been already described (p. 510), whDe for a battery, one or two Leclanchd or sawdust Daniell cells are generally the most convenient The number of cells to be used depends, however, on the magnitude of the resistance to be determined and the sensitiveness of the galvanometer. The key k is inserted in the battery circuit in order that the battery may be thrown out, except when required for the measurement The continual passage of a current through the coils of the box heats them, and if the current be strong enough may do damage. It will be noticed that at each of the points a, b, c, d, three conductors meet, and that including the galvanometer and battery there are six conductors in all, joining the four points a, B, c, D. When the resistances are such that the N N 2 548 Practical Physics. [Ch. XX. § n- current in the conductor joining two of the points is inde- pendent of the E.M.F. in the conductor joining the odier two, then those two conductors are said to be conjugate. In the Wheatstone's bridge method of measuring reast- ances the battery and galvanometer circuits are made to be conjugate ; the current through the galvanometer is inde- pendent of the E.M.F. of the battery. If the equation p/q = r/s hold, the galvanometer is not deflected whatever be the KM.F. of the battery ; there is no need, therefore, to use a constant battery. Moreover, since we only require to determine when no current flows through the galvanometer circuit, and not to measure a steady' current, a sensitive galvanoscope is all that is neces- sary ; we do not need to know the relation between the current and the deflexion produced by it Fig. 66 is another diagram d the connections, which shews more clearly the conjugate rela- tion. The conductors a b and cr ^are conjugate if the equation p/q = r/s holds. It follows from this that we may interchange the galvano- meter and battery without affecting the working of ti« method. The galvanometer may be placed between * and B, and the battery between c and d. The sensitive ness of the measurements will, however, depend on th relative positions of the two, and the following rule is givtc by Maxwell, * Electricity and Magnetism,' voL i. § 348, - determine which of the two arrangements to adopt Of thf two resistances, that of the battery and that of the galvaiKy meter, connect the greater resistance, so as to join the t^ greater to the two less of the other four. As we shall see directly, it will generally happen wha Cu. XX. $ 77.] Ohfris Law. 549 making the final measurements, that Q and 8 are greater than p and r ; thus, referring to fig. 65, the connections are there arranged to suit the case in which the resistance of the battery is greater than that of the galvanometer. To measure a Resistance with the Wluatstone-bridge Box, Make the connections as shewn in fig. 65. Be sure that the binding screws are everywhere tight and that the copper wires are clean and bright at all points where there are contacts. This is especially necessary for the wires which connect r to the box. Any resistance due to them or their contacts will of course be added to the value of r. For delicate measurements contacts must be made by means of thick copper rods amalgamated with mercury, and dipping into mercury cups. The bottoms of the cups should be covered with discs of amalgamated copper, and the wires must press on to these with a steady pressure throughout the experiment; it is not sufficient to make the contact through the mercury by letting the wires drop into it without touching the copper bottom. The cups themselves are conveniently made of pill boxes, covered with a good thick coat of vambh. See that all the plugs are in their places in the box, and press them firmly in with a screw motion to ensure efficient contact Bring the control magnet of the galvanometer down near the coils, and if the resistance to be measured be not even approximately known, it generally saves time to shunt the galvanometer, using the shunt, provided there be one, if not, a piece of thin German-silver wire. Take two equal resistances out of the arms p and Q. Since it is probable that the galvanometer will be somewhat too sensitive even ivhen shunted, it is better to take out the two 100 ohm 3lugs rather than the two 10 ohms. Then, since p = Q, it will be equal to s. Take i ohm out from 8. Make contact first with the 550 Practical Physics. \cvu XX. f n« battery key k^ and then with the galvanometer key k', and note the direction of the deflexion — suppose it be to the right Take out looo ohms from s, and note the deflexion — suppose it be to the left The resistance is clearly between I and looo ohms. Now take out 500 ohms — let the deflexion be to the left — R is less than 500. Proceed thus, and suppose that with 67 ohms the deflexion is to the left, and that with 66 ohms it is to the right The resistance r is clearly between 66 and 67 ohms. Now make p 10 ohms and Q 100, and at the same time remove the shunt, and raise the galvanometer magnet to increase the sensitiveness. Since q is ten times p, s must be ten times r to obtain a balance. Thus s must be between 660 and 670. Supi>ose that it is found that with 665 ohms the deflexion is to the left, and with 664 it is to the right, the true value of s is between 664 and 665, and since R = ps/q, the true value of r is between 66*4 and 66*5. We have thus found a third figure in the value of R. Now make Q 1000 ohms and p 10 ohms. Then, since q is 100 times p, s must be 100 times r to secure the balance; and it will be found that when s b 664a the deflexion is to the right ; when it is 6650 it is to the left The galvano- meter may now be made as sensitive as possible ; and it will probably be found that with a value of s, such as 6646, there is a small deflexion to the right, and with s equal to 6647 a small deflexion to the left Thus the value of r b between 66*46 and 66*47. If the fourth flgure be required correctly, we may find it by interpolation as follows : — When s is 6646 let the deflexion to the right be a scale divisions, and when it is 6647 let it be ^divisions to the left Then since an addition of i ohm to the value ol R alters the reading by a+^ scale divisions, it will require an addition of «/(« +^) ohms to alter it by a divisions. Cb. XX. S 77.] OhnCs Law. 5 5 1 Thus the true value *of r is 6646+0/(^7+^) ohms, and the value of s is 66-46+a/ioo(a+^) ohms. The exactness to which the determination can be carried will depend on the accuracy with which the small out- standing deflexions a and b can be read, and on the con- stancy of the battery. If it be found that the resistance R is less than i ohm, make p 10 ohms, and Q 100 ; then the value of s will be ten times that of r, and if we find that s lies between 5 and 6, it follows that r is between -5 and *6 ; then make p 10 ohms, and q iooo, and proceed similarly. After making the determination the connecting wires must all be removed fi-om the box and the plugs replaced. Experiment — Determine the values of the resistances in the given box. Enter results thus : — Nominal value Real value 10 ohms • • • 10*03 ohms 20 „ • • . 20-052 „ 50 H • • . 50005 „ 100 „ • • . IOOI3 „ Measurement of a Galvanometer Resistance — Thomson* s Method. It has been shewn that if, m the Wheatstone's bridge arrangement, two of the conductors, as ab, cd (fig. tt^ p. 548), are conjugate, then the current through the one due to an E.M.F. in the other is zero. It follows from this that the current through the other conductors is independent of the resistance in c d, and is the same whether c d be con- nected by a conductor or be insulated ; for the condition that the two should be conjugate is that c and d should be at the same potential, and if this condition be satisfied there will never be any tendency for a current to flow along c d ; 552 Practical Physics. [Ch. XX. § Th the currents io the rest of the circutt will, therefore, not depend on c d. Suppose, now, a galvanometer is placed in the branch DA, and a key in CD (fig. 67), there will be a deflexion produced in the galvanometer. Adjust the resistance s until Mr galvanometer deflexion is unal- tered by making or breaking con- tact in the branch cd. When this is the case it follows that A B and c D are conjugate, and, therefore, that p R= - X s. Q But R is the resistance of the galvanometer, which is thus measured by a null method without the use of a second gal- vanometer. Fig. 68 shews the connections, using the Wheatstone- bridge box. A considerable portion of the current from the Fig. 68. battery flows through the galvanometer, and the needle is thereby deflected. If a Thomson's galvanometer be used in the ordinary manner, the spot of light will be quite ofi* the scale. In order to ascertain if the adjustment of the resistances is correct the mirror must be brought back to near its zero position by the aid of permanent magnets ; it is probable that the control magnet will be too weak to do this alone, and others must be employed in addition. This constitutes one of the defects of the method ; the field of magnetic force in which the needle hangs thus bp(U>mes very strong, and the sensi- tiveness of the galvanometer is thus diminished. By using a y^xy weak electromotive force we may dispense with the CH.XX. §770 Ohnis Law. 553 additional magnets ; the control magnet itself may be suffi- cient. We may attain this end by shunting the battery with a German-silver wire. The resistance suitable will depend on many conditions, and must be found by trial A more economical method of diminishing the electro- motive force between the points a and b is to introduce resistance into the battery circuit between point a or b and the pole. By making this interpolated resistance sufficiently great we may make the KM.F. between a and b, what frac- tion we please of the total E.M.F. of the battery. And by increasing the resistance of the circuit we diminish the cur- rent which flows, and therefore diminish the consumption of zinc in the battery, whereas if the E. M. F. between a and b be reduced by shunting, the total current supplied by the battery is increased, and a larger expenditure of zinc is the result The battery used should be one of fairly constant E.M.F., for, if not, the current through the galvanometer will vary, and it will be difficult to make the necessary observations. The method of proceeding is the same as that employed in the last section ; the arms p and Q are first made equal, and two values found, differing by one ohm, between which s lies. The ratio p/q is then made -i, and the first decimal place in the value of r obtained, and so on. Experiment, — Determine, by Thomson's method, the re- sistance of the given galvanometer. Enter result thus : — Galvanometer No. 6 . Resistance 66-3 ohms. Measurement of a Battery Resistance— Manc^s Method. If we recollect that electromotive forces can be super- posed, and that the resultant effect is simply the sum of the individual effects produced by each, it is clear that the con- dition that two conductors in a Wheatstone bridge, such as A B and c D (fig. 66), may be conjugate b not altered by the 554 Practical Physics. fca. XX. $ 77. introduction of a second battery into any of the arms of the bridge. Such a battery will of course send a current through the galvanometer, and produce a deflexion, which will be superposed on that due to the battery in a b. Let a battery be put in the arm a d (fig. 69), r being its resist- ance, and let the galvanometer needle be brought back to its zero position by the use of external magnets. Adjust the resistance s until making or breaking contact in the battery circuit a b produces no effect on the galvanometer ; that is, until the circuits a b and c d are conjugate. When this is the case we have R = p s/q ; and p, s, and Q being known, we can find r, the resistance of the battery. There is, however, no need for a second battery in a b ; for the effect on the galvanometer due to this battery is zero when the conjugate condition is satisfied, whatever be its E.M.F. Take then the case when the E.M.F. is zero, i.e. connect a and b directly through a conductor. If the conjugate condition be satisfied this will produce no effect on the galvanometer ; the deflexion due to the battery in A D will not be altered. Again take the case in which the E.M.F. produced between a and b by the battery in a b is exactly equal and opposite to that produced between those points by the battery in a d. The galvanometer deflexion will still, if the conjugate condition hold, be unaltered. But in this case no current flows along a b ; the conditions are the same as if A and b were insulated. Thus the battery in a B may be supposed removed and replaced by a key. If the resistance s be adjusted until no effect is produced on the galvanometer by making con- tact with this key, it follows that the conjugate conditi<xi holds, and therefore r = ps/q, so that r is determined. I'his is the principle of Mance's method. Ch. XX. f 77.] Ohnis Law, 555 Fig. 69 gives a diagram of the arrangement Fig. 70 shews how the connections are made with the Wheatstone- bridge box. The method of procedure is as follows : — Make the arms p and Q equal. Make contact in the bat- tery circuit with the key k'. Since any resistance which may exist in this key will of necessity be included in the measure- b ment of the resistance r, it is important that its resistance should be small enough to be neglected It is advisable to have a key in the circuit, for, as we have said aheady, it is always best to allow the current to flow through the coUs only when actually re- quired for the experiment Bring the spot of light back to the centre of the scale by the use of the control magnet and, if re- qutsitCy by shunting the gal- vanometer. Determine thus two values of s differing by i ohm, between which r lies. It must be remembered that any variation in s alters the permanent current through the galvanometer, and therefore the control magnet may require readjustment each time s is changed Make the ratio p/q *i and proceed in the same way to find the first decimal place in the value of R. Then make the ratio *oi and find a second decimal. One difficulty requires special notice. It is true that making or breaking contact in the circuit a b will, if the conjugate condition hold, have no direct effect on the current in c d. It does, however, alter the total amount of 556 Practical Physics. [Ch. XX. fn- current which is being produced by the battery. When ad is closed an additional circuit is open for this current ; now with most batteries the E.M.F. depends somewhat on the current which the battery is producing, that is, on the rate at which chemical changes are going on in it ; so that when the battery is called upon to do more work by the closing of the circuit ab, its E.M.F. is gradually altered and the permanent deflexion is thereby changed. On making con- tact with the key the spot of light may move, not because the conjugate condition is not satisfied, but because of this change in the E.M.F. of the battery. This is a funda- mental defect in the method, and prevents the attainment of results of the highest accuracy. The difficulty may be partially obviated as follows : — It will be found that the displacement produced through the conjugate condition not being satisfied is a somewhat sudden jerk, while that which arises from variation in the E.M.F. is more gradual in its nature. A little practice is all that is required to recognise the difference between the two. Now it will always be possible to arrange the resistances so that the two displace- ments are in opposite directions. Let us suppose that it is found that when s is too large on making contact the jeik is to the right ; the gradual deflexion to the left. Gradually decrease s undl the jerk appears to be zero, and the spot seems to move steadily to the left, and take the value of s thus found as the one required The results thus obtained will be found fairly consistent A more exact method for overcoming the difficulty, doe to Professor O. J. Lodge, was described by him in the * Phi- losophical Magazine' of 1876. This, however, involves the use of a specially constructed key, and for an account of it the reader must be referred to the original paper. Experiment, — Determine by Mance's method the resistance of the given battery. Enter results thus : — X Lcclanch^ cell (a) . , . , lai ohm Ch. XX. 1 77.] Oktris Law. 557 I Ledandi^ cell ifi) , • • ix)9 ohm I Sawdust Danidl • • • • 10*95 »» I Cylinder Daniell . • • • -58 ^ 78. The British Attociation Wire Bridge. — ^Keasurement of Electrical Besistanoa The apparatus used for measuring resistances by the Wheatstone-bridge method frequently takes another form. The theory of the method is of course the same as when the box is employed, but instead of varying the resistance s, the ratio p/q is made capable of continuous alteration. The conductors bc, c a of figure 64 are two portions of a straight wire of platinum- silver or German-silver, or some other material of a high specific resistance, which is care- fidly drawn so as to have a uniform cross-section, the re- sistance of any portion of such a wire being proportional to its length. The ratio of the resistances p/q will be the ratio of the two lengths a c/b c A sliding-piece or jockey moves along this wire, and by pressing a spring attached to it electrical connection with the galvanometer can be made at any desired point c of a b. Thus the ratio of a c to b c can be made to have any value by altering the position of the point c along this wu:e. A scale, usually divided to millimetres, is fixed parallel to the wire ; the ends of the wire a and b coincide with the extremities of the scale ; and the position of the point c, at which the contact is made, can be read by means of a mark on the sliding-piece. The ends of this wire are fixed to stout copper pieces, by means of which connection is made with the resistances r and s. These copper strips are so thick that for many purposes their resistance may be neglected when compared with that of the wire a c b. The apparatus usually takes the form shewn in fig. 71. The strips n 11 a, n' m' b are the stout copper pieces just referred to. It will be noticed that there are gaps left between m and a, m' and b ; their purpose will be explained 558 Practical Physics. [Ch. XX. § ?«• shortly (p. 560). When the bridge is used as described above, these two gaps are closed by two strips of copper^ shewn by dotted lines in the figure, which are screwed tightly down to the fixed copper pieces. The wire r, whose resistance is required, and s, the standard, are electrically connected with the apparatus, either by means of binding screws or of mercury cups, as may be most convenient; Fig. 71. tf^ ^t .> M..- N q 1^' wn ^ft.■■^.■■ly■.M..■.lw.■ly.■Wpl■..■W..l.■.■w■ j....i^.j,...w.j....iy...i....m..i..'M 1 eLIk binding screws are also provided for the battery and gal- vanometer wires. To make a determination of the value of r, close the gaps A M and b m' and connect the resistances, battery, and galvanometer, as shewn in the figure. Close the battery circuit by the key k. Move the jockey c until a pod- tioN is found for it, such that no defiexion is produced in the galvanometer on making contact at c. Let a and k be the lengths of the two pieces of the bridge wire on either side of c. Then we have r/s = p/Q = a/^, and R = s a/^ The apparatus may conveniently be used to find the specific resistance of the material of which a wh-e is com- posed. For if R be the resistance, and p the specific re- sistance of a wire of length / and uniform circular cross- Ch. XX. § 78.] OhffCs Law, 5 59 section of diameter d^ then the area of the cross-section \s ^T^, and we have so that The value of r can be found by the method just described. The length of the wire may be measured with a steel tape, or other suitable apparatus, and the diameter d can be determined by the aid of the screw gauge. For great accuracy this method of finding the diameter may not be suffi- cient. It may be more accurately calculated from a know- ledge of the mass, length, and density of the wire (see§ 8). The determination of R by the method just described is not susceptible of very great accuracy. The position of c cannot be found with very great exactness, and an error in this will produce very considerable error in the result It can be shewn as follows that the effect of an error x in the position of c produces least effect in the result when c is the middle point of the wire. For let c be the whole length of the wire ; then we have found that c—a Suppose that an error x has been made in the position of c, so that the true value of a is a-hx Then the true value of R is R+x, say, where <i-¥x c^a—x Hence if we neglect terms involving x^ we have c—a\ a(c—a)) \ a(r--a)> Hence X_ XJ R a{c^a) 56o Practicai Physics. [Ch. XX. § iK Now it is shewn in books on Algebra that a{f^(£) is greatest when a = c—a^ that is, when a = ^^ or c is at the middle point of the bridge-wire ; and in this case the ratio of x to r, that is, the ratio of the error produced by an eiror a: in a to the resistance measured, is least when c is at the middle point Thus the standard chosen for s should have approximatelj the same value as r. This may be conveniently arranged for by using a resistance-box for s and taking out plugs untO the adjusted position of c is near the middle of the wire. But even with this precaution the method is far frcHo sensitive ; the resistance of the wire n n' is probably vwy small compared with the resistances R and s. Nearly aD the current flows directly through the wire, and very litdc through the coils r and s. The greatest possible difiference of potential between c and d is small, and the deflexion of the galvanometer will always be small. To remedy this two other resistance coils are inserted m the gaps AM and bm', the copper strips being removed Suppose their resistances respectively are p' and q', and suppose that the value of r is known approximately, or has been found from a rough observation as above. The values of p', q' must be such the ratio of p' to q' does not difiier much from that of R to s. Suppose that when the position ot equilibrium is found the lengths of wire on either side of c are a and ^, and that the resistance of a unit length of the wire is known to be ir. Then, if we neglect the resistances of the copper strips ic h and m'n' — these will be exceedingly small, and may be neglected without sensible error — the value of p will be p'-i-<j<r, and that of o« ^*'\'bv^ and we have R_PjM^ S Q'+ba The value of r is thus determined, and it can be shewn that the error in the result produced by a given error in the position of c is much less than when there is no resistance between a and m, b and m'. Ch. XX. § 78.] OhnCs Law. 561 This method involves a knowledge of <y, the resistance of a centimetre of the bridge-wire. To find this the resistance of the whole wire may be measured with a Post-Office box, or otherwise, and the result divided by the length of the wire in centimetres. Another method of determining o- will be given in the next section. Moreover, since a a- and Bkt are small compared with p' and q', it follows that, as stated above, the ratio r/s must not differ much from the ratio p'/q'. Experiments. (i) Measure by means of a resistance box and the wire bridge the resistance of the given coils. (2) Determine accurately the length of the given wire which has a resistance of i ohm. (3) Determine also the specific resistance of the material of the wire. Enter results thus : — (1} Nominal values Obserred Talnea I ohm. . . 1*013 ohm* 10 „ . . . IO'22 ^ 20 „ . . . 2018 (2) Length of wire given, 250 cm. P'- I ohm. S - I „ a -43*2 b -56-8 <r « '001 8 ohm. :, R - -5129 „ Length of wire having a resistance of i ohm = 487*4 cm. (3) Same wire used as in (2). Diameter (mean of ten observations with screw gauge) «-*i2 11 cm. Specific resistance, 23,640 abs. units per cm. cube. = 23640 X lo'^ ohms per cm. cube. 79. Carey Fetter's Kethod of Comparing Besistances. The B.A. wire bridge just described is most useful when it is required to determine the difference between two o o _nJTTB 562 Practical Physics. [Ch- XX. 1 79- nearly equal resistances of from one to ten ohms in value The method of doing this, which is due to Professor Caiejr Foster, is as follows. Let r and s be the two nearly equal resistances to be com- pared; p and Q two other nearly equal re- sistances, which shooM, to give the greatest ac- curacy, not differ much from R and s. We do not require to know anything abcmt p and Q except that they are nearly equaL It is convenient to have them wound together on the same bobbin, for then we can be sure that Chey are always at the same temperature. Place R and s in the gaps am, b m' of the bridge, and p and Q in the gaps a d and d b respectively. Let a and h^ as before, be the lengths of the bridge-wire on either side erf c when the galvanometer needle is in equilibrium. Let x, v be the unknown resistances of the two strips m n and m'n'. Fig-' 72 shews the arrangement Then, if o- be the resistance of one centimetre of the bridge- wire, we have P_R + x-fdt<r , X — — ;; ■ ' > — • • • • III Interchange the po- sition of R and s and determine another po- sition c (fig. 73), for the galvanometer contact in which there is no "n deflexion. Let a', ^' be the corresponding vi> lues of a and b. Thes Q R+Y+2V . (t) Ch. XX. § 79.] Ohm's Law. 563 And by adding unity to each side we have, from equations (i)and(2) R + X + go-+S-f Y-f 3(r _ P + Q S + Y+^<r Q "rTT+^V • ^^^ Also d(+^ = whole length of bridge wire = a'+^' . . (4) .-. R+X+tf<r+S + Y + ^o- = S + X-|-a'a- + R + Y + ^V . (5) Hence from (3) S + Y+^<r = R+Y + ^<r; .% R— s = (^-^)o- = (a'-a)<r, by (4). .(6) Now (a'— a)<r is the resistance of a portion of the bridge wire equal in length to the distance through which the sliding-piece has been moved. This distance can be measured with very great accuracy, and thus the difference of the resistances of the two coils can be very exactly deter- mined. To obtain all the accuracy of which the method is capable, it is necessary that the contacts should be good, and should remain in the same condition throughout Mercury cups should generally be employed to make con- tact, and it is necessary that the electrodes of the various coils should be pressed firmly on to the bottoms of these either by weights, or, if convenient, by means of spring clamps. At the three points c, n, n', we have contacts of two dissimilar metals. These points are probably at different temperatures — the observei^s hand at c tends to raise its temperature — and a difference of temperature in a circuit of different metals will, it is known, produce a thermo- electric current in the circuit This current will, under the circumstances of the expenment, be very small ; still, it may be a source of error. 002 5^4 Practical Physics. [Ch. XX. $ 79. The best method of getting rid of its effects is to place a commutator in the battery circuit, and make two obsenra- tions of each of the lengths a and af^ reversing the batteiy between the twa It can be shewn that the mean of dte two observations gives a value free from the error produced by the thermo-electric effect Again, a variation in the temperature of a conductor produces an alteration in its resistance. For very accurate work it is necessary to keep the coils r and s at known tem- peratures. This is generally done by means of a water-batfa, in which the coils are immersed. It has been found that for most of the metals, at any rate within ordinary limits of temperature, the change of resistance per degree of temperature is very nearly constant, so that if R be the resistance of a coil at temperature /°C, Ro its resistance at o% and a the coefficient of increase of resistance per degree of temperature, we have R = Ro(l+a/). Carey Foster's method is admirably adapted for finding this quantity a. The standard coil s is kept at one definite temperature, and the values of the difference between its resistance and that of the other coil are observed for two tem- peratures of the latter. Let these temperatures be /| and tf, and the corresponding resistances R| and R^; then we have 0=(Ri«Ra)/Ro(/i-/2). The observations have given us the values of R| — s and R^ -s with great accuracy, and fix>m them we can get Rj— R, ; an approximate value of Ro will be all that is required for our purpose, for it will be found that a is a veiy small quantity, and we have seen (p. 44) that we may with- out serious error employ an approximate value in the de- nominator of a small fraction. Whenever precautions are requisite to maintain the coils at a uniform temperature, the interchanging of tbe Cb. XX. § 79*1 Ohm* 5 Law. 565 coils R, s is a source of difficulty with the ordinary arrange ments. Time is lost in moving the water-jackets in which the coils are immersed, and the temperature may vary. The contacts, moreover, are troublesome to adjust. To obviate this, among other difficulties, a special form of bridge was devised by Dr. J. A. Fleming, and described in the * Pro ceedings of the Physical Society of London,' vol iii. The ordinary bridge may be easily adapted to an arrangement similar to Fleming's, as follows, egfh (Rg. 74) are four mercury cups ; b and f Fic. 74* are connected by stout copper rods with a and M, o and H with b and m' respectively. For the first obser- vation the electrodes of R are placed in s and r being held in their position by weights or spring clamps, while the electrodes of s are in g and h. For the second observation the electrodes of r arc placed in o and h, those of s in e and f, as shewn by the dotted lines. This interchange is easily effected. The water jackets need not be displaced ; the coils can readily be moved in them. The connections ae, mf, &c., may conveniently be made of stout copper rod, fastened down to a board of dry wood, coated with paraffin. To make the mercury cups the ends of these rods are tumed up through a right angle and cut off level They are then amalgamated and short pieces of india-rubber tubing are slipped over them, and tied round with thin wire ; the india-rubber tubing projects above the rod, and thus forms the cup. The other ends of the rods are made to fit the binding screws of the ordinary bridge.^ * For a fuller account of this and other similar contrivances, tec Pkilosophiaa Magmnt, May 1884. 566 Practical Physics. [Ch. XX. f 7> Calibration of a Bri^e-wire. The method gives us also the best means of calibrating a bridge-wire. Make an observation exactly as above Alter the value of p slightly by inserting in series with it a short piece of German-silver wire. The only effect will be to shift somewhat the positions of c and c' along the scale, and thus the difference between r and s is obtained in terms of the length of a different part of the bridge-wire. If the wire be of uniform section the two lengths thus obtained will be the same. If they are not the same, it follows that the area of the cross-section, or the specific resistance of the wire, is different at different points, and a table of conections can be formed as for a thermometa (p. 242). If the difference between the two coils be accurately known we can determine from the observations the value of the resistance of a centimetre of the bridge-wire. This is given by equation (6) ; for the values of R— s and «*— « ve known, and we have <r=(R— s)/(a'— <i). For this purpose the following method is often con- venient. Take two i-ohm coils and place in multiple arc with one of them a lo-ohm coil Let the equivalent re- sistance of this combination be R ; then the value of r is 10/ 1 1 ohms. Instead of interchanging the coils place the ten in multiple arc with the other single ohm and make the observation as before ; then in this case we have r — 8= I = — ohm. II XJ and if / be the distance through which the jockey has been moved we obtain ^ = 122221, CH. XX. 8 79.] OhnCs Law, 567 ExperimenU. (i) Calibrate the bridge-wire. (2) Determine the average resistance of one centimetre of it (3) Determine accurately the difference between the resist- ance of the given coil and the standard i-ohm at the tempera- ture of the room. Enter results thus : — (i) Value of R-s for calibration, '009901 — being the differ- ence between i ohm and i ohm with 100 in multiple arc — PoddoD of c Value of a' - « Division 20 . . • . 5*48 ,,40. . , . 549 „ 60. . . 551 9, 80. . . 5*52 (2) R— S « 09091 ohm. / (mean of 5 observations) « 50*51 cm. <r«'ooi79ohm. (3) Difference between the given coil and the standard at temperature of IS^'C, observed three times. Values *oo37, x»36, '00372 ohm. Mean •00367 olun. 80. FoggendorfTfl Method for the CompariBon of Electro- motive Forces. Latimer-Clark^s Potentiometer. The method given in § 76 for the comparison of electro- motive forces is subject to a defect similar to that men- tioned in § 77, on the measurement of resistance ; that is, it depends upon measuring the deflexion of a galvano- meter needle, and assumes that the KM.F. of the batteries employed remain constant throughout the experiment The following method, first suggested by Poggendorff, resembles the Wheatstone-bridge method for measuring resistances, in being a null method ; it depends, that is to say, on determining when no current passes through a gal* vanometer, not on measuring the deflexion. We have seep 568 Practical Physics. [Ch. XX. § aa (p. 528) that if a current c be flowing through a conductor, the E.M.F. or difference of potential between any two points^ separated by a resistance r, is or. Let A B (fig. 75) be a conductor of considerable resist* ance, through which a current is flowing from a to b ; let Pi be a point on this conductor, E| the difference of potential between a and P}. If a and P] be connected by a second wire A Gi P|, including a galvanometer Oi in its circuit, a current will flow from a to Pi through this wire also. L^ a second battery be placed in this circuit in such a way as to tend to produce a current in the direction P| G| A|; the cur- rent actually flowing through the galvanometer Oj will depend on Uie difference between Ei and the E.M.F. of this Fig. 7$. battery. By varying the position of Pj along the wire a b, we can adjust matters so that no current flows through the galvanometer Oj ; when this is the case it is clear that the E.M.F. E, of the battery is equal to the difference of poten- tial between a and P| produced by the first battery. Let the resistance aPj be R|, and let R be the resistance of a b, and p that of the battery which is producing the current through A B, including, of course, any connecting wires, b being the RM.F. of this battery. Then, if c be the current in a b, we have E,=cR,=ERi/(R+p) (p. 528), or. •1 B+P Ch. XX. § 80.] Ohtris Law. 569 lliis equation gives us, if we know p, the ratio Bi/e; for R and Ri can be observed. This method will be satisfactory in practice if r is very great compared with p, for then an approximate value of p will be sufficient ; or if r is sufficiently large, p may be entirely neglected, and we may write E|/e = R|/r. This is Poggendorffs method of comparing the KM.F. of two batteries The following arrangement, suggested by Latimer-Clark, obviates the necessity for knowing pt Let E], B) be the two KM.F. to be compared, e that of a third battery, producing a current between the two points A and b; e must be greater than Ei or E3. Connect the three positive poles of the three batteries to a, the negative pole of E to B, and the negative poles of Si and Ej, through two galvanometers Gj and G], to two points Pi, Pj on ab; adjust the positions of P] and p^ separately until no current flows through either galvanometer. It will be found con- venient to have two keys, Kj, K2, in the circuits for the pur- poses of this adjustment Thus, positions are to be found for pj and P2, such that on making contact simultaneously with the two keys there is no deflexion observed in either galvanometer. Let R], r^ be the resistances of a Pi, a p, respectively, when this is the case. Then, c being the cur- rent in A B, we have E| ss C Rj, E2 ^— C Rj* . Ei_Ri • . —^ "^ • Eq R2 By this method of procedure results are obtained en- tirely independent of the battery used to give the main current through a b. The differences of potential actually compared are those between the two poles of the batteries respectively, when neither is producing a current A convenient experimental arrangement for carrying out the comparison of electromotive forces on this method 570 Practical Physics. [Ch. XX. § te as described by LAtimer-Clark, has been called a *• potentio- meter.' The use of the t